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  • 1. Acker, A.
    et al.
    Henrot, A.
    Poghosyan, M.
    Shahgholian, Henrik
    KTH, Superseded Departments, Mathematics.
    The multi-layer free boundary problem for the p-Laplacian in convex domains2004In: Interfaces and free boundaries (Print), ISSN 1463-9963, E-ISSN 1463-9971, Vol. 6, no 1, p. 81-103Article in journal (Refereed)
    Abstract [en]

    The main result of this paper concerns existence of classical solutions to the multi-layer Bernoulli free boundary problem with nonlinear joining conditions and the p-Laplacian as governing operator. The present treatment of the two-layer case involves technical refinements of the one-layer case, studied earlier by two of the authors. The existence treatment of the multi-layer case is largely based on a reduction to the two-layer case, in which uniform separation of the free boundaries plays a key role.

  • 2. Acker, Andrew
    et al.
    Poghosyan, Michael
    Shahgholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Convex configurations for solutions to semilinear elliptic problems in convex rings2006In: Communications in Partial Differential Equations, ISSN 0360-5302, E-ISSN 1532-4133, Vol. 31, no 9, p. 1273-1287Article in journal (Refereed)
    Abstract [en]

    For a given convex ring Omega = Omega(2)\(Omega) over bar (1) and an L-1 function f : Omega x R -> R+ we show, under suitable assumptions on f, that there exists a solution (in the weak sense) to Delta(p)u = f(x, u) in Omega u = 0 on partial derivative Omega(2) u = M on partial derivative Omega(1) with {x is an element of Omega : u(x) > s} boolean OR Omega(1) convex, for all s is an element of (0, M).

  • 3.
    Aleksanyan, Hayk
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Shahgholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Discrete balayage and boundary sandpile2016In: Journal d'Analyse Mathematique, ISSN 0021-7670, E-ISSN 1565-8538Article in journal (Refereed)
  • 4.
    Aleksanyan, Hayk
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Shahgholian, Henrik
    Perturbed divisible sandpiles and quadrature surfaces2017Article in journal (Refereed)
  • 5.
    Aleksanyan, Hayk
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). The University of Edinburgh.
    Shahgholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Sjölin, Per
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Applications of Fourier analysis in homogenization of Dirichlet problem I. Pointwise estimates2013In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 254, no 6, p. 2626-2637Article in journal (Refereed)
    Abstract [en]

    In this paper we prove convergence results for homogenization problem for solutions of partial differential system with rapidly oscillating Dirichlet data. Our method is based on analysis of oscillatory integrals. In the uniformly convex and smooth domain, and smooth operator and boundary data, we prove pointwise convergence results, namely vertical bar u(epsilon)(x) - u(0)(x)vertical bar <= C-kappa epsilon((d-1)/2) 1/d(x)(kappa), for all x is an element of D, for all kappa > d - 1, where u(epsilon) and u(0) are solutions of respectively oscillating and homogenized Dirichlet problems, and d(x) is the distance of x from the boundary of D. As a corollary for all 1 <= p < infinity we obtain L-P convergence rate as well.

  • 6.
    Aleksanyan, Hayk
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). The University of Edinburgh.
    Shahgholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Sjölin, Per
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Applications of Fourier Analysis in Homogenization of Dirichlet Problem III: Polygonal Domains2014In: Journal of Fourier Analysis and Applications, ISSN 1069-5869, E-ISSN 1531-5851, Vol. 20, no 3, p. 524-546Article in journal (Refereed)
    Abstract [en]

    In this paper we prove convergence results for the homogenization of the Dirichlet problem for elliptic equations in divergence form with rapidly oscillating boundary data and non oscillating coefficients in convex polygonal domains. Our analysis is based on integral representation of solutions. Under a certain Diophantine condition on the boundary of the domain and smooth coefficients we prove pointwise, as well as convergence results. For larger exponents we prove that the convergence rate is close to optimal. We also suggest several directions of possible generalization of the results in this paper.

  • 7.
    Aleksanyan, Hayk
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). The University of Edinburgh.
    Shahgholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Sjölin, Per
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Applications of Fourier Analysis in Homogenization of the Dirichlet Problem: L-p Estimates2015In: Archive for Rational Mechanics and Analysis, ISSN 0003-9527, E-ISSN 1432-0673, Vol. 215, no 1, p. 65-87Article in journal (Refereed)
    Abstract [en]

    Let u(epsilon) be a solution to the system div(A(epsilon)(x)del u(epsilon)(x)) = 0 in D, u(epsilon)(x) = g(x, x/epsilon) on partial derivative D, where D subset of R-d (d >= 2), is a smooth uniformly convex domain, and g is 1-periodic in its second variable, and both A(epsilon) and g are sufficiently smooth. Our results in this paper are twofold. First we prove L-p convergence results for solutions of the above system and for the non-oscillating operator A(epsilon)(x) = A(x), with the following convergence rate for all 1 <= p < infinity parallel to u(epsilon) - u(0)parallel to (LP(D)) <= C-P {epsilon(1/2p), d = 2, (epsilon vertical bar ln epsilon vertical bar)(1/p), d = 3, epsilon(1/p), d >= 4, which we prove is (generically) sharp for d >= 4. Here u(0) is the solution to the averaging problem. Second, combining our method with the recent results due to Kenig, Lin and Shen (Commun Pure Appl Math 67(8): 1219-1262, 2014), we prove (for certain class of operators and when d >= 3) ||u(epsilon) - u(0)||(Lp(D)) <= C-p[epsilon(ln(1/epsilon))(2)](1/p) for both the oscillating operator and boundary data. For this case, we take A(epsilon) = A(x/epsilon), where A is 1-periodic as well. Some further applications of the method to the homogenization of the Neumann problem with oscillating boundary data are also considered.

  • 8.
    Aleksanyan, Hayk
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). The University of Edinburgh.
    Shahgholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Sjölin, Per
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    L2-estimates for singular oscillatory integral operators2016In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 441, no 2, p. 529-548Article in journal (Refereed)
    Abstract [en]

    In this note we study singular oscillatory integrals with linear phase function over hypersurfaces which may oscillate, and prove estimates of L2L2 type for the operator, as well as for the corresponding maximal function. If the hypersurface is flat, we consider a particular class of a nonlinear phase functions, and apply our analysis to the eigenvalue problem associated with the Helmholtz equation in R3.

  • 9. Andersson, John
    et al.
    Lindgren, Erik
    Shahgholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Optimal Regularity for the No-Sign Obstacle Problem2013In: Communications on Pure and Applied Mathematics, ISSN 0010-3640, E-ISSN 1097-0312, Vol. 66, no 2, p. 245-262Article in journal (Refereed)
    Abstract [en]

    In this paper we prove the optimal C-1,C-1(B-1/2)-regularity for a general obstacle-type problem Delta u = f chi({u not equal 0}) in B-1, under the assumption that f * N is C-1,C-1(B-1), where N is the Newtonian potential. This is the weakest assumption for which one can hope to get C-1,C-1-regularity. As a by-product of the C-1,C-1-regularity we are able to prove that, under a standard thickness assumption on the zero set close to a free boundary point x(0), the free boundary is locally a C-1-graph close to x(0) provided f is Dini. This completely settles the question of the optimal regularity of this problem, which has been the focus of much attention during the last two decades.

  • 10.
    Andersson, John
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Lindgren, Erik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Shahgholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Optimal regularity for the obstacle problem for the p-Laplacian2015In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 259, no 6, p. 2167-2179Article in journal (Refereed)
    Abstract [en]

    In this paper we discuss the obstacle problem for the p-Laplace operator. We prove optimal growth results for the solution. Of particular interest is the point-wise regularity of the solution at free boundary points. The most surprising result we prove is the one for the p-obstacle problem: Find the smallest u such thatdiv(|∇u|p-2∇u)≤0,u≥ϕ,in B1, with ϕ∈C1,1(B1) and given boundary datum on ∂B1. We prove that the solution is uniformly C1,1 at free boundary points. Similar results are obtained in the case of an inhomogeneity belonging to L∞. When applied to the corresponding parabolic problem, these results imply that any solution which is Lipschitz in time is C1,1p-1 in the spatial variables.

  • 11.
    Andersson, John
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Lindgren, Erik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Shahgholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Optimal regularity for the parabolic no-sign obstacle type problem2013In: Interfaces and free boundaries (Print), ISSN 1463-9963, E-ISSN 1463-9971, Vol. 15, no 4, p. 477-499Article in journal (Refereed)
    Abstract [en]

    We study the parabolic free boundary problem of obstacle type Delta u - partial derivative u/partial derivative t = f chi({u not equal 0}). Under the condition that f = H nu for some function nu with bounded second order spatial derivatives and bounded first order time derivative, we establish the same regularity for the solution u. Both the regularity and the assumptions are optimal. Using this result and assuming that f is Dini continuous, we prove that the free boundary is, near so called low energy points, a C-1 graph. Our result completes the theory for this type of problems for the heat operator.

  • 12. Andersson, John
    et al.
    Shah Gholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Weiss, Georg S.
    The singular set of higher dimensional unstable obstacle type problems2013In: Rendiconti Lincei - Matematica e Applicazioni, ISSN 1120-6330, E-ISSN 1720-0768, Vol. 24, no 1, p. 123-146Article in journal (Refereed)
  • 13.
    Andersson, John
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Shahgholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Global solutions of the obstacle problem in half-spaces, and their impact on local stability2005In: Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, E-ISSN 1432-0835, Vol. 23, no 3, p. 271-279Article in journal (Refereed)
    Abstract [en]

    We show that there are an abundance of non-homogeneous global solutions to the obstacle problem, in the half-space, Delta u = X-{u > 0}, u >= 0 inR(+)(2), with a (fixed) homogeneous boundary condition u(0, x(2)) = lambda(2) (x(2)(+))(2) (0 < lambda < 1/root 2) As a consequence we obtain local instability of the free boundary under C-1,C-1 perturbation, of the Dirichlet data.

  • 14.
    Andersson, John
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Shahgholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Uraltseva, Nina N.
    Weiss, Georg S.
    Equilibrium points of a singular cooperative system with free boundary2015In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 280, p. 743-771Article in journal (Refereed)
    Abstract [en]

    In this paper we initiate the study of maps minimising the energy integral(D)(vertical bar del u vertical bar(2) + 2 vertical bar u vertical bar) dx, which, due to Lipschitz character of the integrand, gives rise to the singular Euler equations Delta u = u / vertical bar u vertical bar chi({vertical bar u vertical bar >0}), u = (u(1,) ... ,u(m)) Our primary goal in this paper is to set up a road map for future developments of the theory related to such energy minimising maps. Our results here concern regularity of the solution as well as that of the free boundary. They are achieved by using monotonicity formulas and epiperimetric inequalities, in combination with geometric analysis.

  • 15. Andersson, John
    et al.
    Shahgholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Weiss, Georg S.
    Double Obstacle Problems with Obstacles Given by Non-C-2 Hamilton-Jacobi Equations2012In: Archive for Rational Mechanics and Analysis, ISSN 0003-9527, E-ISSN 1432-0673, Vol. 206, no 3, p. 779-819Article in journal (Refereed)
    Abstract [en]

    We prove optimal regularity for double obstacle problems when obstacles are given by solutions to Hamilton-Jacobi equations that are not C (2). When the Hamilton-Jacobi equation is not C (2) then the standard Bernstein technique fails and we lose the usual semi-concavity estimates. Using a non-homogeneous scaling (different speeds in different directions) we develop a new pointwise regularity theory for Hamilton-Jacobi equations at points where the solution touches the obstacle. A consequence of our result is that C (1)-solutions to the Hamilton-Jacobi equation <Equation ID="Equa"> <MediaObject> </MediaObject> </Equation>, are, in fact, C (1,alpha/2), provided that . This result is optimal and, to the authors' best knowledge, new.

  • 16.
    Andersson, John
    et al.
    Mathematics Institute, University of Warwick.
    Shahgholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Weiss, Georg S.
    Mathematical Institute of the Heinrich Heine University.
    On the singularities of a free boundary through Fourier expansion2012In: Inventiones Mathematicae, ISSN 0020-9910, E-ISSN 1432-1297, Vol. 187, no 3, p. 535-587Article in journal (Refereed)
    Abstract [en]

    In this paper we are concerned with singular points of solutions to the unstable free boundary problem Delta u = -chi({u>0}) in B-1. The problem arises in applications such as solid combustion, composite membranes, climatology and fluid dynamics. It is known that solutions to the above problem may exhibit singularities-that is points at which the second derivatives of the solution are unbounded-as well as degenerate points. This causes breakdown of by-now classical techniques. Here we introduce new ideas based on Fourier expansion of the non-linearity chi({u>0}). The method turns out to have enough momentum to accomplish a complete description of the structure of the singular set in R-3. A surprising fact in R-3 is that although u(rx)/sup(B1) vertical bar u(rx)vertical bar can converge at singularities to each of the harmonic polynomials xy, x(2) + y(2)/2 - z(2) and z(2) - x(2) + y(2)/2, it may not converge to any of the non-axially-symmetric harmonic polynomials alpha((1 + delta)x(2) + (1 - delta)y(2) - 2z(2)) with delta not equal 1/2. We also prove the existence of stable singularities in R-3.

  • 17. Andersson, John
    et al.
    Shahgholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Weiss, Georg S.
    Uniform Regularity Close to Cross Singularities in an Unstable Free Boundary Problem2010In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 296, no 1, p. 251-270Article in journal (Refereed)
    Abstract [en]

    We introduce a new method for the analysis of singularities in the unstable problem Delta u = chi{u> 0}, which arises in solid combustion as well as in the composite membrane problem. Our study is confined to points of "supercharacteristic" growth of the solution, i.e. points at which the solution grows faster than the characteristic/invariant scaling of the equation would suggest. At such points the classical theory is doomed to fail, due to incompatibility of the invariant scaling of the equation and the scaling of the solution. In the case of two dimensions our result shows that in a neighborhood of the set at which the second derivatives of u are unbounded, the level set {u = 0} consists of two C-1-curves meeting at right angles. It is important that our result is not confined to the minimal solution of the equation but holds for all solutions.

  • 18. Apushkinskaya, D. E.
    et al.
    Ural’Tseva, N. N.
    Shahgholian, Henrik
    KTH, Superseded Departments, Mathematics.
    Lipschitz property of the free boundary in the parabolic obstacle problem2004In: St. Petersburg Mathematical Journal, ISSN 1061-0022, E-ISSN 1547-7371, Vol. 15, no 3, p. 375-391Article in journal (Refereed)
    Abstract [en]

    A parabolic obstacle problem with zero constraint is considered. It is proved, without any additional assumptions on a free boundary, that near the fixed boundary where the homogeneous Dirichlet condition is fulfilled, the boundary of the noncoincidence set is the graph of a Lipschitz function.

  • 19. Arakelyan, A.
    et al.
    Shah Gholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Multi-Phase Quadrature Domains and a Related Minimization Problem2016In: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929X, p. 1-21Article in journal (Refereed)
    Abstract [en]

    In this paper we introduce the multi-phase version of the so-called Quadrature Domains (QD), which refers to a generalized type of mean value property for harmonic functions. The well-established and developed theory of one-phase QD was recently generalized to a two-phase version, by one of the current authors (in collaboration). Here we introduce the concept of the multi-phase version of the problem, and prove existence as well as several properties of such solutions. In particular, we discuss possibilities of multi-junction points.

  • 20.
    Arnarson, Teitur
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Djehiche, Boualem
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Poghosyan, Michael
    Yerevan State Univ, Dept Math & Mech.
    Shahgholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    A PDE approach to regularity of solutions to finite horizon optimal switching problems2009In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 71, no 12, p. 6054-6067Article in journal (Refereed)
    Abstract [en]

    We study optimal 2-switching and n-switching problems and the corresponding system of variational inequalities. We obtain results on the existence of viscosity solutions for the 2-switching problem for various setups when the cost of switching is non-deterministic. For the n-switching problem we obtain regularity results for the solutions of the variational inequalities. The solutions are C-l,C-l-regular away for the free boundaries of the action sets.

  • 21. Blank, Ivan
    et al.
    Shahgholian, Henrik
    KTH, Superseded Departments, Mathematics.
    Boundary Regularity and Compactness for Overdetermined Problems2003In: Annali della Scuola Normale Superiore di Pisa (Classe Scienze), Serie V, ISSN 0391-173X, E-ISSN 2036-2145, Vol. 2, no 4, p. 787-802Article in journal (Refereed)
    Abstract [en]

    Let D be either the unit ball B-1(0) or the half ball B-1(+)(0), let f be a strictly positive and continuous function, and let u and Omega subset of D solve the following overdetermined problem: Delta u (x) = chi(Omega) (x) f (x) in D, 0 is an element of partial derivative Omega, u = vertical bar del u vertical bar = 0 in Omega(c), where chi(Omega) denotes the characteristic function of Omega, Omega(c) denotes the set D\Omega, and the equation is satisfied in the sense of distributions. When D = B-1(+)(0), then we impose in addition that u(x) 0 {(x', x(n))vertical bar x(n) = 0} We show that a fairly mild thickness assumption on Omega(c) will ensure enough compactness on it to give us "blow-up" limits, and we show how this compactness leads to regularity of partial derivative Omega. In the case where f is positive and Lipschitz, the methods developed in Caffarelli, Karp, and Shahgholian (2000) lead to regularity of partial derivative Omega under a weaker thickness assumption.

  • 22. Caffarelli, L. A.
    et al.
    Karp, L.
    Shahgholian, Henrik.
    KTH, Superseded Departments, Mathematics.
    Regularity of a free boundary with application to the Pompeiu problem2000In: Annals of Mathematics, ISSN 0003-486X, E-ISSN 1939-8980, Vol. 151, no 1, p. 269-292Article in journal (Refereed)
    Abstract [en]

    In the unit ball B(0, 1), let u and Omega (a domain in R-N) salve the following overdetermined problem: Delta u = chi(Omega) in B(0, 1), 0 is an element of partial derivative Omega, u = \del u\ = 0 in B(0, 1) \ Omega, where chi(Omega) denotes the characteristic function, and the equation is satisfied in the sense of distributions. If the complement of Omega does not develop cusp singularities at the origin then we prove partial derivative Omega is analytic in some small neighborhood of the origin. The result can be modified to yield for more general divergence form operators. As an application of this, then, we obtain the regularity of the boundary of a domain without the Pompeiu property, provided its complement has no cusp singularities.

  • 23. Caffarelli, L.
    et al.
    Petrosyan, A.
    Shahgholian, Henrik
    KTH, Superseded Departments, Mathematics.
    Regularity of a free boundary in parabolic potential theory2004In: Journal of The American Mathematical Society, ISSN 0894-0347, E-ISSN 1088-6834, Vol. 17, no 4, p. 827-869Article in journal (Refereed)
  • 24. Caffarelli, L.
    et al.
    Salazar, J.
    Shahgholian, Henrik.
    KTH, Superseded Departments, Mathematics.
    Free-boundary regularity for a problem arising in superconductivity2004In: Archive for Rational Mechanics and Analysis, ISSN 0003-9527, E-ISSN 1432-0673, Vol. 171, no 1, p. 115-128Article in journal (Refereed)
    Abstract [en]

    This paper concerns regularity properties of the mean-field theory of superconductivity. The problem is reminiscent of the one studied earlier by L.A. Caffarelli, L. Karp and H. Shahgholian in connection with potential theory. The difficulty introduced in this paper is the existence of several patches, where on each patch the solution to the problem may have different constant values. However, using a refined analysis, we reduce the problem to the one-patch case, at least locally near ''regular'' free boundary points. Using a monotonicity formula, due to Georg S. Weiss, we characterize global solutions of a related equation. Hence earlier regularity results apply and we conclude the C-1 regularity of the free boundary.

  • 25. Caffarelli, Luis A.
    et al.
    Shahgholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Regularity of free boundaries a heuristic retro2015In: Philosophical Transactions. Series A: Mathematical, physical, and engineering science, ISSN 1364-503X, E-ISSN 1471-2962, Vol. 373, no 2050, article id 20150209Article, review/survey (Refereed)
    Abstract [en]

    This survey concerns regularity theory of a few free boundary problems that have been developed in the past half a century. Our intention is to bring up different ideas and techniques that constitute the fundamentals of the theory. We shall discuss four different problems, where approaches are somewhat different in each case. Nevertheless, these problems can be divided into two groups: (i) obstacle and thin obstacle problem; (ii) minimal surfaces, and cavitation flow of a perfect fluid. In each case, we shall only discuss the methodology and approaches, giving basic ideas and tools that have been specifically designed and tailored for that particular problem. The survey is kept at a heuristic level with mainly geometric interpretation of the techniques and situations in hand.

  • 26.
    Caffarelli, Luis A.
    et al.
    Univ Texas Austin, Dept Math, Austin, TX 78712 USA..
    Shahgholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Yeressian, Karen
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    A MINIMIZATION PROBLEM WITH FREE BOUNDARY RELATED TO A COOPERATIVE SYSTEM2018In: Duke mathematical journal, ISSN 0012-7094, E-ISSN 1547-7398, Vol. 167, no 10, p. 1825-1882Article in journal (Refereed)
    Abstract [en]

    We study the minimum problem for the functional integral(Omega)(vertical bar del u vertical bar(2) + Q(2) chi({vertical bar u vertical bar>0}))dx with the constraint u(i) >= 0 for i = 1,... , m, where Omega subset of R-n is a bounded domain and u = (u(1),... , u(m)) is an element of H-1 (Omega;R-m). First we derive the Euler equation satisfied by each component. Then we show that the noncoincidence set {vertical bar u vertical bar > 0} is (locally) nontangentially accessible. Having this, we are able to establish sufficient regularity of the force term appearing in the Euler equations and derive the regularity of the free boundary Omega boolean AND partial derivative{vertical bar u vertical bar> 0}.

  • 27. Chen, Gui-Qiang
    et al.
    Shahgholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Vazquez, Juan-Luis
    Free boundary problems: the forefront of current and future developments2015In: Philosophical Transactions. Series A: Mathematical, physical, and engineering science, ISSN 1364-503X, E-ISSN 1471-2962, Vol. 373, no 2050, article id 20140285Article in journal (Other academic)
  • 28. Danielli, D.
    et al.
    Petrosyan, A.
    Shahgholian, Henrik.
    KTH, Superseded Departments, Mathematics.
    A singular perturbation problem for the p-Laplace operator2003In: Indiana University Mathematics Journal, ISSN 0022-2518, E-ISSN 1943-5258, Vol. 52, no 2, p. 457-476Article in journal (Refereed)
    Abstract [en]

    In this paper we initiate the study of the nonlinear one phase singular perturbation problem div(\delu(epsilon)\(p-2)delu(epsilon)) = beta(epsilon)(u(epsilon)), (1 < p < infinity) in a domain Omega of R-N. We prove uniform Lipschitz regularity of uniformly bounded solutions. Once this is done we can pass to the limit to obtain a solution to the stationary case of a combustion problem with a nonlinearity of power type. (The case p = 2 has been considered earlier by several authors.).

  • 29. de Queiroz, Olivaine S.
    et al.
    Shahgholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    A free boundary problem with log term singularity2017In: Interfaces and free boundaries (Print), ISSN 1463-9963, E-ISSN 1463-9971, Vol. 19, no 3, p. 351-369Article in journal (Refereed)
    Abstract [en]

    We study a minimum problem for a non-differentiable functional whose reaction term does not have scaling properties. Specifically we consider the functional (sic)(v) = integral(Omega) (vertical bar del v vertical bar(2)/2 - v(+)(log v - 1))dx -> min which should be minimized in some natural admissible class of non-negative functions. Here, v(+) = max{0, v}. The Euler-Lagrange equation associated with (sic) is -Delta u = chi({u>0}) log u, which becomes singular along the free boundary partial derivative{u > O}. Therefore, the regularity results do not follow from classical methods. Besides, the logarithmic forcing term does not have scaling properties, which are very important in the study of free boundary theory. Despite these difficulties, we obtain optimal regularity of a minimizer and show that, close to every free boundary point, they exhibit a super-characteristic growth like r(2)vertical bar log r vertical bar. This estimate is crucial in the study of analytic and geometric properties of the free boundary.

  • 30. Emamizadeh, Behrouz
    et al.
    Prajapat, Jyotshana V.
    Shahgholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    A Two Phase Free Boundary Problem Related to Quadrature Domains2011In: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929X, Vol. 34, no 2, p. 119-138Article in journal (Refereed)
    Abstract [en]

    In this paper we introduce a two phase version of the well-known Quadrature Domain theory, which is a generalized (sub)mean value property for (sub)harmonic functions. In concrete terms, and after reformulation into its PDE version the problem boils down to finding solution to -Delta u=(mu(+)-lambda(+))(chi{u>0}) - (mu(-) - lambda(-))(chi{u<0}) in IRN. where lambda(+/-) >0 are given constants and mu(+/-), are non-negative bounded Radon measures, with compact support. Our primary concern is to discuss existence and geometric properties of solutions.

  • 31. Figalli, Alessio
    et al.
    Shahgholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    A General Class of Free Boundary Problems for Fully Nonlinear Elliptic Equations2014In: Archive for Rational Mechanics and Analysis, ISSN 0003-9527, E-ISSN 1432-0673, Vol. 213, no 1, p. 269-286Article in journal (Refereed)
    Abstract [en]

    In this paper we study the fully nonlinear free boundary problem {F(D(2)u) = 1 almost everywhere in B-1 boolean AND Omega vertical bar D(2)u vertical bar <= K almost everywhhere in B-1\Omega, where K > 0, and Omega is an unknown open set. Our main result is the optimal regularity for solutions to this problem: namely, we prove that W (2,n) solutions are locally C (1,1) inside B (1). Under the extra condition that and a uniform thickness assumption on the coincidence set {D u = 0}, we also show local regularity for the free boundary partial derivative Omega boolean AND B-1.

  • 32. Figalli, Alessio
    et al.
    Shahgholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    A general class of free boundary problems for fully nonlinear parabolic equations2015In: Annali di Matematica Pura ed Applicata, ISSN 0373-3114, E-ISSN 1618-1891, Vol. 194, no 4, p. 1123-1134Article in journal (Refereed)
    Abstract [en]

    In this paper, we consider the fully nonlinear parabolic free boundary problem { F(D(2)u) - partial derivative(t)u = 1 a.e. in Q(1) boolean AND Omega vertical bar D(2)u vertical bar + vertical bar partial derivative(t)u vertical bar <= K a.e. in Q(1)\Omega, where K > 0 is a positive constant, and Omega is an (unknown) open set. Our main result is the optimal regularity for solutions to this problem: namely, we prove that W-x(2,) (n) boolean AND W-t(1,) (n) solutions are locally C-x(1,) (1) boolean AND C-t(0,) (1) inside Q(1). A key starting point for this result is a new BMO-type estimate, which extends to the parabolic setting the main result in Caffarelli and Huang (Duke Math J 118(1): 1-17, 2003). Once optimal regularity for u is obtained, we also show regularity for the free boundary partial derivative Omega boolean AND Q(1) under the extra condition that Omega superset of{u not equal 0}, and a uniform thickness assumption on the coincidence set {u = 0}.

  • 33. Figalli, Alessio
    et al.
    Shahgholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    An overview of unconstrained free boundary problems2015In: Philosophical Transactions. Series A: Mathematical, physical, and engineering science, ISSN 1364-503X, E-ISSN 1471-2962, Vol. 373, no 2050, article id 20140281Article in journal (Refereed)
    Abstract [en]

    In this paper, we present a survey concerning unconstrained free boundary problems of type F-1(D(2)u, del u, u, x) = 0 in B-1 boolean AND Omega, F-2(D(2)u, del u, u, x) = 0 in B-1 \ Omega, u is an element of S(B-1), where B-1 is the unit ball, Omega is an unknown open set, F-1 and F-2 are elliptic operators (admitting regular solutions), and S is a functions space to be specified in each case. Our main objective is to discuss a unifying approach to the optimal regularity of solutions to the above matching problems, and list several open problems in this direction.

  • 34. Fotouhi, M.
    et al.
    Shahgholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    A semilinear PDE with free boundary2017In: Nonlinear Analysis, Theory, Methods and Applications, ISSN 0362-546X, Vol. 151, p. 145-163Article in journal (Refereed)
    Abstract [en]

    We study the semilinear problem Δu=λ+(x)(u+)q−1−λ−(x)(u−)q−1inB1, from a regularity point of view for solutions and the free boundary ∂{±u&gt;0}. Here B1 is the unit ball, 1&lt;q&lt;2 and λ± are Lipschitz. Our main results concern local regularity analysis of solutions and their free boundaries. One of the main difficulties encountered in studying this equation is classification of global solutions. In dimension two we are able to present a fairly good analysis of global homogeneous solutions, and hence a better understanding of the behavior of the free boundary. In higher dimensions the problem becomes quite complicated, but we are still able to state partial results; e.g. we prove that if a solution is close to one-dimensional solution in a small ball, then in an even smaller ball the free boundary can be represented locally as two C1-regular graphs Γ+=∂{u&gt;0} and Γ−=∂{u&lt;0}, tangential to each other. It is noteworthy that the above problem (in contrast to the case q=1) introduces interesting and quite challenging features, that are not encountered in the case q=1. E.g. one obtains homogeneous global solutions that are not one-dimensional. This complicates the analysis of the free boundary substantially.

  • 35. Hakobyan, A
    et al.
    Shahgholian, Henrik
    KTH, Superseded Departments, Mathematics.
    A uniqueness result for an overdetermined problem in non-linear parabolic potential theory2004In: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929X, Vol. 21, no 4, p. 405-414Article in journal (Refereed)
    Abstract [en]

    In this paper we study the question of uniqueness for an inverse problem, arising in the (thermal) linear and/or non-linear potential theory. The overdetermined problem we shall study is represented by (div(\delu\(p-2)delu) - D(t)u - chi(Omega) + mu)u = 0, where supp(mu) subset of Omega subset of R-n x (0,infinity), 1 < p < infinity, mu is an element of L-infinity, and Omega boolean AND {t = tau} is bounded for tau > 0. The problem has applications in shape-recognition in underground water/oil recovery, subject to shape-change during time intervals. The particular case u greater than or equal to 0, D(t)u greater than or equal to 0, and p = 2, is an example of the well-known Stefan.

  • 36. Henrot, A.
    et al.
    Shahgholian, Henrik.
    KTH, Superseded Departments, Mathematics.
    Existence of classical solutions to a free boundary problem for the p-Laplace operator: (I) the exterior convex case2000In: Journal für die Reine und Angewandte Mathematik, ISSN 0075-4102, E-ISSN 1435-5345, Vol. 521, p. 85-97Article in journal (Refereed)
    Abstract [en]

    In this paper we prove, under convexity assumptions for the data, the existence of classical solutions for a Bernoulli-type free boundary problem, with the p-Laplacian as the governing operator. The method employed here originates from a pioneering work of A. Beurling where he proves the existence for the harmonic case in the plane, though with no geometrical restrictions.

  • 37. Henrot, A.
    et al.
    Shahgholian, Henrik.
    KTH, Superseded Departments, Mathematics.
    Existence of classical solutions to a free boundary problem for the p-Laplace operator: (II) The interior convex case2000In: Indiana University Mathematics Journal, ISSN 0022-2518, E-ISSN 1943-5258, Vol. 49, no 1, p. 311-323Article in journal (Refereed)
    Abstract [en]

    In this paper, we prove the existence of convex classical solutions for a Bernoulli-type free boundary problem, in the interior of a convex domain. The governing operator considered is the p-Laplacian. This work is inspired by the pioneering work of A. Beurling where he proves the existence for the harmonic case in the plane, using the notion of sub- and super-solutions in a geometrical sense.

  • 38. Henrot, A.
    et al.
    Shahgholian, Henrik.
    KTH, Superseded Departments, Mathematics.
    The one phase free boundary problem for the p-Laplacian with non-constant Bernoulli boundary condition2002In: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850, Vol. 354, no 6, p. 2399-2416Article in journal (Refereed)
    Abstract [en]

    Our objective, here, is to generalize our earlier results on the existence of classical convex solution to a free boundary problem with a Bernoulli-type boundary gradient condition and with the p-Laplacian as the governing operator. The main theorems of this paper assert that the exterior and the interior free boundary problem with a Bernoulli law, i.e. with a prescribed pressure a(x) on the free streamline of the flow, have convex solutions provided the initial domains are convex. The continuous function a(x) is subject to certain convexity properties. In our earlier results we have considered the case of constant a(x). In the lines of the proof of the main results we also prove the semi-continuity (up to the boundary) of the gradient of the p-capacitary potentials in convex rings, with C-1 boundaries.

  • 39. Karakhanyan, A. L.
    et al.
    Kenig, C. E.
    Shahgholian, Henrik.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    The behavior of the free boundary near the fixed boundary for a minimization problem2007In: Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, E-ISSN 1432-0835, Vol. 28, no 1, p. 15-31Article in journal (Refereed)
    Abstract [en]

    We show that the free boundary partial derivative{u > 0}, arising from the minimizer(s) u, of the functional J(u) = (Omega)integral vertical bar del u vertical bar(2) + lambda(2)(+)chi{u > 0} + lambda(2)(-)chi{u < 0}. approaches the (smooth) fixed boundary a Omega tangentially, at points where the Dirichlet data vanishes along with its gradient.

  • 40. Karakhanyan, A. L.
    et al.
    Shahgholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Analysis of a free boundary at contact points with Lipschitz data2015In: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850, Vol. 367, no 7, p. 5141-5175Article in journal (Refereed)
    Abstract [en]

    In this paper we consider a minimization problem for the functional J(u) = ∫ B+<inf>1</inf> |∇u|2 + λ2 <inf>+</inf>χ<inf>{u>0}</inf> + λ2 <inf>−</inf>χ<inf>{u≤0}</inf> in the upper half ball B+ <inf>1</inf> ⊂ ℝn, n ≥ 2, subject to a Lipschitz continuous Dirichlet data on ∂B+ <inf>1</inf>. More precisely we assume that 0 ∈ ∂{u > 0} and the derivative of the boundary data has a jump discontinuity. If 0 ∈ ∂({u > 0} ∩ B+ <inf>1</inf>), then (for n = 2 or n ≥ 3 and the one-phase case) we prove, among other things, that the free boundary ∂{u > 0} approaches the origin along one of the two possible planes given by γx<inf>1</inf> = ±x<inf>2</inf>, where γ is an explicit constant given by the boundary data and λ± the constants seen in the definition of J(u). Moreover the speed of the approach to γx<inf>1</inf> = x<inf>2</inf> is uniform.

  • 41. Karakhanyan, A. L.
    et al.
    Shahgholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Boundary behaviour for a singular perturbation problem2016In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 138, p. 176-188Article in journal (Refereed)
    Abstract [en]

    In this paper we study the boundary behaviour of the family of solutions (uε) to singular perturbation problem δuε=βε(uε),(divides)uε(divides)≤1 in B1+=(xn>0)∩((divides)x(divides)<1), where a smooth boundary data f is prescribed on the flat portion of ∂B1+. Here βε((dot operator))=1εβ((dot operator)ε),β∈C0∞(0,1),β≥0,∫01β(t)=M>0 is an approximation of identity. If ∇f(z)=0 whenever f(z)=0 then the level sets ∂(uε>0) approach the fixed boundary in tangential fashion with uniform speed. The methods we employ here use delicate analysis of local solutions, along with elaborated version of the so-called monotonicity formulas and classification of global profiles.

  • 42. Karp, L.
    et al.
    Kilpelainen, T.
    Petrosyan, A.
    Shahgholian, Henrik.
    KTH, Superseded Departments, Mathematics.
    On the porosity of free boundaries in degenerate variational inequalities2000In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 164, no 1, p. 110-117Article in journal (Refereed)
    Abstract [en]

    In this note we consider a certain degenerate variational problem with constraint identically zero. The exact growth of the solution near th free boundary is established. A consequence of this is that the free boundary is porous and therefore its Hausdorff dimension is less than N and hence it is of Lebesgue measure zero.

  • 43. Karp, L.
    et al.
    Shahgholian, Henrik.
    KTH, Superseded Departments, Mathematics.
    Regularity of a free boundary at the infinity point2000In: Communications in Partial Differential Equations, ISSN 0360-5302, E-ISSN 1532-4133, Vol. 25, no 12-Nov, p. 2055-2086Article in journal (Refereed)
    Abstract [en]

    Suppose there is a nonnegative function u and an open set Ohm subset of R-n(n greater than or equal to 3), satisfying Deltau = chi (Ohm) in B-r(e), u = \delu\ = 0 on B-r(e)\Ohm, where B-r(e) = {x : \x\ > r}. Under a certain thickness condition on R-n\Ohm, we prove that the boundary of {x:x/\x\(2) is an element of Ohm} is a graph of a C-1 function in a neighborhood of the origin. As a by-product of the method of the proof, we also obtain the following result: Replace chi (Ohm) by f chi (Ohm), with a certain assumptions on f. Then for any solution u which is asymptotically nonnegative at infinity, there holds lim(r-->infinity) (\Br\)/(\Ohm boolean AND Br\) is an element of {1/2,1}.

  • 44. Kawohl, B.
    et al.
    Shahgholian, Henrik.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Gamma limits in some Bernoulli free boundary problem2005In: Archiv der Mathematik, ISSN 0003-889X, E-ISSN 1420-8938, Vol. 84, no 1, p. 79-87Article in journal (Refereed)
    Abstract [en]

    We study the limit cases p --> infinity and p --> 1 of the functionals (1) E-p(u) := integral(Rn) {1/p(|delu|/a) (p) + p-1/ p chi({u>0})} dx, where u = 1 on a given compact set K subset of R-n, and a > 0 is also given. Minimizers u(p) of these functionals have uniformly bounded support Omega(p) := {u(p) > 0} and satisfy (2) - Delta(p)u(p) = 0 in Omega(p), u(p) = 1 on K, |delu(p)| = a on partial derivativeOmega(p).

  • 45. Kilpelaeinen, Tero
    et al.
    Shahgholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Zhong, Xiao
    Growth estimates through scaling for quasilinear partial differential equations2007In: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, E-ISSN 1798-2383, Vol. 32, no 2, p. 595-599Article in journal (Refereed)
    Abstract [en]

    In this note we use a scaling or blow up argument to obtain estimates to solutions of equations of p-Laplacian type.

  • 46. Kim, S.
    et al.
    Lee, K. -A
    Shahgholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    An elliptic free boundary arising from the jump of conductivity2017In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 161, p. 1-29Article in journal (Refereed)
    Abstract [en]

    In this paper we consider a quasilinear elliptic PDE, div(A(x,u)∇u)=0, where the underlying physical problem gives rise to a jump for the conductivity A(x,u), across a level surface for u. Our analysis concerns Lipschitz regularity for the solution u, and the regularity of the level surfaces, where A(x,u) has a jump and the solution u does not degenerate. In proving Lipschitz regularity of solutions, we introduce a new and unexpected type of ACF-monotonicity formula with two different operators, that might be of independent interest, and surely can be applied in other related situations. The proof of the monotonicity formula is done through careful computations, and (as a byproduct) a slight generalization to a specific type of variable matrix-valued conductivity is presented.

  • 47. Lee, K. A.
    et al.
    Shahgholian, Henrik
    KTH, Superseded Departments, Mathematics.
    Regularity of a free boundary for viscosity solutions of nonlinear elliptic equations2001In: Communications on Pure and Applied Mathematics, ISSN 0010-3640, E-ISSN 1097-0312, Vol. 54, no 1, p. 43-56Article in journal (Refereed)
    Abstract [en]

    Our objective in this paper is to analyze the regularity of the boundary of a set Omega2 that admits a solution u to the following overdetermined problem: F(D(2)u) = chi (Omega) in B, u = \ delu \ = 0 in B \ Omega. Here F is a convex, uniformly elliptic operator of homogeneity one, B is the unit ball in R-n, and the equation is satisfied in the viscosity sense. We also consider the case of concave F in R-2.

  • 48. Lee, K.
    et al.
    Shahgholian, Henrik
    KTH, Superseded Departments, Mathematics.
    Hausdorff measure and stability for the p-obstacle problem (2 < p 2003In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 195, no 1, p. 14-24Article in journal (Refereed)
    Abstract [en]

    In this paper we analyze the free boundary for the inhomogeneous obstacle problem with zero obstacle governed by the degenerate operator div(A/(delu)) = fchi({u>0}), where f is a positive, Lipschitz function, and A(delu) is of the p-Laplacian type, i.e., A(delu) approximate to \delu\(p-2)delu, (2

  • 49.
    Lindgren, Erik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Shahgholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Edquist, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    On the two-phase membrane problem with coefficients below the Lipschitz threshold2009In: Annales de l'Institut Henri Poincare. Analyse non linéar, ISSN 0294-1449, E-ISSN 1873-1430, Vol. 26, no 6, p. 2359-2372Article in journal (Refereed)
    Abstract [en]

    We study the regularity of the two-phase membrane problem, with coefficients below the Lipschitz threshold. For the Lipschitz coefficient case one can apply a monotonicity formula to prove the C-1,C-1-regularity of the solution and that the free boundary is, near the so-called branching points, the union of two C-1-graphs. In our case, the same monotonicity formula does not apply in the same way. In the absence of a monotonicity formula, we use a specific scaling argument combined with the classification of certain global solutions to obtain C-1,C-1-estimates. Then we exploit some stability properties with respect to the coefficients to prove that the free boundary is the union of two Reifenberg vanishing sets near so-called branching points.

  • 50. Manfredi, J.
    et al.
    Petrosyan, A.
    Shahgholian, Henrik
    KTH, Superseded Departments, Mathematics.
    A free boundary problem for infinity-Laplace equation2002In: Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, E-ISSN 1432-0835, Vol. 14, no 3, p. 359-384Article in journal (Refereed)
    Abstract [en]

    We consider a free boundary problem for the p-Laplacian Delta(mu)u = div(\delu\(p-2)delu), describing nonlinear potential flow past a convex profile K with prescribed pressure \delu(x)\ = a(x) on the free stream line. The main purpose of this paper is to study the limit as p --> infinity of the classical solutions of the problem above, existing under certain convexity assumptions on a(x). We show, as one can expect, that the limit solves the corresponding potential flow problem for the infinity-Laplacian Deltax u = del(2) u delu . delu. in a certain weak sense, strong enough however, to guarantee uniqueness. We show also that in the special case a(x) = a(0) > 0 the limit is given by the distance function.

12 1 - 50 of 74
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