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Publications (10 of 69) Show all publications
Caffarelli, L. A., Shahgholian, H. & Yeressian, K. (2018). A MINIMIZATION PROBLEM WITH FREE BOUNDARY RELATED TO A COOPERATIVE SYSTEM. Duke mathematical journal, 167(10), 1825-1882
Open this publication in new window or tab >>A MINIMIZATION PROBLEM WITH FREE BOUNDARY RELATED TO A COOPERATIVE SYSTEM
2018 (English)In: Duke mathematical journal, ISSN 0012-7094, E-ISSN 1547-7398, Vol. 167, no 10, p. 1825-1882Article in journal (Refereed) Published
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}.

Place, publisher, year, edition, pages
Duke University Press, 2018
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-232769 (URN)10.1215/00127094-2018-0007 (DOI)000438689000002 ()
Note

QC 20180803

Available from: 2018-08-03 Created: 2018-08-03 Last updated: 2018-08-03Bibliographically approved
de Queiroz, O. S. & Shahgholian, H. (2017). A free boundary problem with log term singularity. Interfaces and free boundaries (Print), 19(3), 351-369
Open this publication in new window or tab >>A free boundary problem with log term singularity
2017 (English)In: Interfaces and free boundaries (Print), ISSN 1463-9963, E-ISSN 1463-9971, Vol. 19, no 3, p. 351-369Article in journal (Refereed) Published
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.

Keywords
Free boundary, regularity theory, logarithmic singularity, porosity
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-220303 (URN)10.4171/IFB/385 (DOI)000416727600002 ()2-s2.0-85031497391 (Scopus ID)
Funder
Swedish Research Council
Note

QC 20171221

Available from: 2017-12-21 Created: 2017-12-21 Last updated: 2017-12-21Bibliographically approved
Kim, S., Lee, K.-A. -. & Shahgholian, H. (2017). An elliptic free boundary arising from the jump of conductivity. Nonlinear Analysis, 161, 1-29
Open this publication in new window or tab >>An elliptic free boundary arising from the jump of conductivity
2017 (English)In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 161, p. 1-29Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
Elsevier Ltd, 2017
Keywords
Conductivity jump, Free boundary problem, Quasilinear elliptic equation, Mathematical techniques, Nonlinear analysis, Free boundary, Free-boundary problems, Lipschitz regularity, Matrix-valued, Monotonicity, Quasi-linear elliptic, Quasilinear elliptic equations, Linear equations
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:kth:diva-216189 (URN)10.1016/j.na.2017.05.010 (DOI)000407182400001 ()2-s2.0-85020887578 (Scopus ID)
Note

QC 20171220

Available from: 2017-12-20 Created: 2017-12-20 Last updated: 2017-12-20Bibliographically approved
Karakhanyan, A. L. & Shahgholian, H. (2016). Boundary behaviour for a singular perturbation problem. Nonlinear Analysis, 138, 176-188
Open this publication in new window or tab >>Boundary behaviour for a singular perturbation problem
2016 (English)In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 138, p. 176-188Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
Elsevier, 2016
Keywords
Free boundary problem, Regularity, Contact points
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:kth:diva-187141 (URN)10.1016/j.na.2015.12.024 (DOI)000374009200010 ()2-s2.0-84955297833 (Scopus ID)
Funder
Swedish Research Council
Note

QC 20160517

Available from: 2016-05-17 Created: 2016-05-17 Last updated: 2017-11-30Bibliographically approved
Aleksanyan, H., Shahgholian, H. & Sjölin, P. (2016). L2-estimates for singular oscillatory integral operators. Journal of Mathematical Analysis and Applications, 441(2), 529-548
Open this publication in new window or tab >>L2-estimates for singular oscillatory integral operators
2016 (English)In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 441, no 2, p. 529-548Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
Elsevier, 2016
Keywords
Helmholtz equation, Maximal operator, Oscillating surface, Singular integral
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:kth:diva-187381 (URN)10.1016/j.jmaa.2016.04.031 (DOI)000375635300002 ()2-s2.0-84963859665 (Scopus ID)
Note

QC 20160527

Available from: 2016-05-27 Created: 2016-05-23 Last updated: 2017-06-20Bibliographically approved
Shah gholian, H. (2016). REGULARITY ISSUES FOR SEMILINEAR PDE-S (A NARRATIVE APPROACH). St. Petersburg Mathematical Journal, 27(3), 577-587
Open this publication in new window or tab >>REGULARITY ISSUES FOR SEMILINEAR PDE-S (A NARRATIVE APPROACH)
2016 (English)In: St. Petersburg Mathematical Journal, ISSN 1061-0022, E-ISSN 1547-7371, Vol. 27, no 3, p. 577-587Article in journal (Refereed) Published
Abstract [en]

Occasionally, solutions of semilinear equations have better (local) regularity properties than the linear ones if the equation is independent of space (and time) variables. The simplest example, treated by the current author, was that the solutions of Delta u = f(u), with the mere assumption that f ' >= -C, have bounded second derivatives. In this paper, some aspects of semilinear problems are discussed, with the hope to provoke a study of this type of problems from an optimal regularity point of view. It is noteworthy that the above result has so far been undisclosed for linear second order operators, with Holder coefficients. Also, the regularity of level sets of solutions as well as related quasilinear problems are discussed. Several seemingly plausible open problems that might be worthwhile are proposed.

Place, publisher, year, edition, pages
American Mathematical Society (AMS), 2016
Keywords
Pointwise regularity, Laplace equation, divergence type equations, free boundary problems
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-186599 (URN)10.1090/spmj/1405 (DOI)000373930300015 ()2-s2.0-84963541486 (Scopus ID)
Note

QC 20160531

Available from: 2016-05-31 Created: 2016-05-13 Last updated: 2017-11-30Bibliographically approved
Figalli, A. & Shahgholian, H. (2015). A general class of free boundary problems for fully nonlinear parabolic equations. Annali di Matematica Pura ed Applicata, 194(4), 1123-1134
Open this publication in new window or tab >>A general class of free boundary problems for fully nonlinear parabolic equations
2015 (English)In: Annali di Matematica Pura ed Applicata, ISSN 0373-3114, E-ISSN 1618-1891, Vol. 194, no 4, p. 1123-1134Article in journal (Refereed) Published
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}.

Keywords
Free boundaries, Regularity, Parabolic fully nonlinear
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-173446 (URN)10.1007/s10231-014-0413-7 (DOI)000359804400009 ()2-s2.0-84931568511 (Scopus ID)
Note

QC 20150915

Available from: 2015-09-15 Created: 2015-09-11 Last updated: 2017-12-04Bibliographically approved
Figalli, A. & Shahgholian, H. (2015). An overview of unconstrained free boundary problems. Philosophical Transactions. Series A: Mathematical, physical, and engineering science, 373(2050), Article ID 20140281.
Open this publication in new window or tab >>An overview of unconstrained free boundary problems
2015 (English)In: 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) Published
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.

Keywords
free boundary problems, obstacle-type, optimal regularity, matching data
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-173962 (URN)10.1098/rsta.2014.0281 (DOI)000360825300008 ()2-s2.0-84939155428 (Scopus ID)
Funder
Swedish Research Council
Note

QC 20151006

Available from: 2015-10-06 Created: 2015-09-24 Last updated: 2017-12-01Bibliographically approved
Karakhanyan, A. L. & Shahgholian, H. (2015). Analysis of a free boundary at contact points with Lipschitz data. Transactions of the American Mathematical Society, 367(7), 5141-5175
Open this publication in new window or tab >>Analysis of a free boundary at contact points with Lipschitz data
2015 (English)In: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850, Vol. 367, no 7, p. 5141-5175Article in journal (Refereed) Published
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.

Keywords
Contact points, Free boundary problem, Regularity
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-167776 (URN)10.1090/S0002-9947-2015-06187-X (DOI)000357045700022 ()2-s2.0-84927613989 (Scopus ID)
Note

QC 20150526

Available from: 2015-05-26 Created: 2015-05-22 Last updated: 2017-12-04Bibliographically approved
Aleksanyan, H., Shahgholian, H. & Sjölin, P. (2015). Applications of Fourier Analysis in Homogenization of the Dirichlet Problem: L-p Estimates. Archive for Rational Mechanics and Analysis, 215(1), 65-87
Open this publication in new window or tab >>Applications of Fourier Analysis in Homogenization of the Dirichlet Problem: L-p Estimates
2015 (English)In: Archive for Rational Mechanics and Analysis, ISSN 0003-9527, E-ISSN 1432-0673, Vol. 215, no 1, p. 65-87Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
Springer, 2015
Keywords
Elliptic-Systems, Green
National Category
Other Mathematics
Identifiers
urn:nbn:se:kth:diva-159614 (URN)10.1007/s00205-014-0774-5 (DOI)000347403500002 ()
Funder
Swedish Research Council
Note

QC 20150209

Available from: 2015-02-09 Created: 2015-02-05 Last updated: 2017-06-20Bibliographically approved
Organisations
Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0002-1316-7913

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