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Publications (10 of 81) Show all publications
Allen, M. & Shahgholian, H. (2019). A New Boundary Harnack Principle (Equations with Right Hand Side). Archive for Rational Mechanics and Analysis, 234(3), 1413-1444
Open this publication in new window or tab >>A New Boundary Harnack Principle (Equations with Right Hand Side)
2019 (English)In: Archive for Rational Mechanics and Analysis, ISSN 0003-9527, E-ISSN 1432-0673, Vol. 234, no 3, p. 1413-1444Article in journal (Refereed) Published
Abstract [en]

We introduce a new boundary Harnack principle in Lipschitz domains for equations with a right hand side. Our approach, which uses comparisons and blow-ups, will adapt to more general domains as well as other types of operators. We prove the principle for divergence form elliptic equations with lower order terms including zero order terms. The inclusion of a zero order term appears to be new even in the absence of a right hand side.

Place, publisher, year, edition, pages
SPRINGER, 2019
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-262754 (URN)10.1007/s00205-019-01415-3 (DOI)000487789700010 ()2-s2.0-85069655650 (Scopus ID)
Note

QC 20191023

Available from: 2019-10-23 Created: 2019-10-23 Last updated: 2019-10-23Bibliographically approved
Aleksanyan, H. & Shahgholian, H. (2019). Discrete balayage and boundary sandpile. Journal d'Analyse Mathematique, 138(1), 361-403
Open this publication in new window or tab >>Discrete balayage and boundary sandpile
2019 (English)In: Journal d'Analyse Mathematique, ISSN 0021-7670, E-ISSN 1565-8538, Vol. 138, no 1, p. 361-403Article in journal (Refereed) Published
Abstract [en]

We introduce a new lattice growth model, which we call the boundary sandpile. The model amounts to potential-theoretic redistribution of a given initial mass on Z(d) (d >= 2) onto the boundary of an (a priori) unknown domain. The latter evolves through sandpile dynamics, and has the property that the mass on the boundary is forced to stay below a prescribed threshold. Since finding the domain is part of the problem, the redistribution process is a discrete model of a free boundary problem, whose continuum limit is yet to be understood. We prove general results concerning our model. These include canonical representation of the model in terms of the smallest super-solution among a certain class of functions, uniform Lipschitz regularity of the scaled odometer function, and hence the convergence of a subsequence of the odometer and the visited sites, discrete symmetry properties, as well as directional monotonicity of the odometer function. The latter (in part) implies the Lipschitz regularity of the free boundary of the sandpile.

As a direct application of some of the methods developed in this paper, combined with earlier results on the classical abelian sandpile, we show that the boundary of the scaling limit of an abelian sandpile is locally a Lipschitz graph.

Place, publisher, year, edition, pages
Springer, 2019
Keywords
Boundary sandpile, balayage, lattice growth model, quadrature surface, divisible sandpile, asymptotic shape, free boundary, Abelian sandpile
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-209237 (URN)10.1007/s11854-019-0037-3 (DOI)000490559900014 ()2-s2.0-85068839050 (Scopus ID)
Note

QCR 20170620 QC 20191115

Available from: 2017-06-17 Created: 2017-06-17 Last updated: 2019-11-15Bibliographically approved
Ghergu, M., Kim, S. & Shahgholian, H. (2019). Exact behavior around isolated singularity for semilinear elliptic equations with a log-type nonlinearity. ADVANCES IN NONLINEAR ANALYSIS, 8(1), 995-1003
Open this publication in new window or tab >>Exact behavior around isolated singularity for semilinear elliptic equations with a log-type nonlinearity
2019 (English)In: ADVANCES IN NONLINEAR ANALYSIS, ISSN 2191-9496, Vol. 8, no 1, p. 995-1003Article in journal (Refereed) Published
Abstract [en]

We study the semilinear elliptic equation -Delta u = u(alpha)vertical bar log u vertical bar(beta) in B-1 \ {0}, where B-1 subset of R-n, with n >= 3, n/n-2 < alpha <n+2/n-2 and -infinity < beta < infinity. Our main result establishes that the nonnegative solution u is an element of C-2(B-1 \ {0}) of the above equation either has a removable singularity at the origin or it behaves like u(x) = A(1 + o(1))vertical bar x vertical bar(-2/alpha-1)(log 1/vertical bar x vertical bar)(-beta/alpha-1) as x -> 0, with A = [(2/alpha-1)(1-beta)(n - 2 - 2/alpha-1)](1/alpha-1).

Place, publisher, year, edition, pages
WALTER DE GRUYTER GMBH, 2019
Keywords
Singular solutions, asymptotic behavior, log-type nonlinearity
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-246290 (URN)10.1515/anona-2017-0261 (DOI)000459891200051 ()2-s2.0-85048763020 (Scopus ID)
Note

QC 20190325

Available from: 2019-03-25 Created: 2019-03-25 Last updated: 2019-03-25Bibliographically approved
Kim, S. & Shahgholian, H. (2019). Homogenization of a Singular Perturbation Problem. Journal of Mathematical Sciences, 242(1), 163-176
Open this publication in new window or tab >>Homogenization of a Singular Perturbation Problem
2019 (English)In: Journal of Mathematical Sciences, ISSN 1072-3374, E-ISSN 1573-8795, Vol. 242, no 1, p. 163-176Article in journal (Refereed) Published
Abstract [en]

We discuss homogenization of the singular perturbation problem Δpuδε=fεβδ(uδε)inℝn\B1¯ with a constant boundary value on the ball. Here, Δp is the usual p-Laplacian operator. It is generally understood that the two parameters δ and ε are in competition and two different behaviors may be exhibited, depending on which parameter tends to zero faster. We consider one scenario where we assume that ε, the homogenization parameter, tends to zero faster than δ, the singular perturbation parameter. We show that there is a universal speed for which the limit solves a standard Bernoulli free boundary problem.

Place, publisher, year, edition, pages
Springer-Verlag New York, 2019
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-263539 (URN)10.1007/s10958-019-04472-x (DOI)2-s2.0-85071018313 (Scopus ID)
Note

QC 20191128

Available from: 2019-11-28 Created: 2019-11-28 Last updated: 2019-12-04Bibliographically approved
Kim, S., Lee, K.-A. & Shahgholian, H. (2019). HOMOGENIZATION OF THE BOUNDARY VALUE FOR THE DIRICHLET PROBLEM. Discrete and Continuous Dynamical Systems, 39(12), 6843-6864
Open this publication in new window or tab >>HOMOGENIZATION OF THE BOUNDARY VALUE FOR THE DIRICHLET PROBLEM
2019 (English)In: Discrete and Continuous Dynamical Systems, ISSN 1078-0947, E-ISSN 1553-5231, Vol. 39, no 12, p. 6843-6864Article in journal (Refereed) Published
Abstract [en]

In this paper, we give a mathematically rigorous proof of the averaging behavior of oscillatory surface integrals. Based on ergodic theory, we find a general geometric condition which we call irrational direction dense condition, abbreviated as IDDC, under which the averaging takes place. It should be stressed that IDDC does not imply any control on the curvature of the given surface. As an application, we prove homogenization for elliptic systems with Dirichlet boundary data, in C-1-domains.

Place, publisher, year, edition, pages
AMER INST MATHEMATICAL SCIENCES-AIMS, 2019
Keywords
Homogenization, boundary layer, oscillatory surface integral, irrational direction, corrector
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-262758 (URN)10.3934/dcds.2019234 (DOI)000488206900005 ()2-s2.0-85072931186 (Scopus ID)
Note

QC 20191023

Available from: 2019-10-23 Created: 2019-10-23 Last updated: 2019-10-23Bibliographically approved
Kim, S., Lee, K.-A. & Shahgholian, H. (2019). Nodal Sets for "Broken" Quasilinear PDEs. Indiana University Mathematics Journal, 68(4), 1113-1148
Open this publication in new window or tab >>Nodal Sets for "Broken" Quasilinear PDEs
2019 (English)In: Indiana University Mathematics Journal, ISSN 0022-2518, E-ISSN 1943-5258, Vol. 68, no 4, p. 1113-1148Article in journal (Refereed) Published
Abstract [en]

We study the local behavior of the nodal sets of the solutions to elliptic quasilinear equations with nonlinear conductivity part, div(A(s) (x, u)del u) = div (f) over bar (x), where A(s) (x, u) has "broken" derivatives of orders s >= 0, such as A(s)(x, u) = a(x) + b(x) (u(+))(s), with (u(+))(0) being understood as the characteristic function on {u > 0}. The vector (f) over bar (x) is assumed to be C-alpha in case s = 0, and C-1,C-alpha (or higher) in case s > 0. Using geometric methods, we prove almost complete results (in analogy with standard PDEs) concerning the behavior of the nodal sets. More precisely, we show that the nodal sets, where solutions have (linear) nondegeneracy, are locally smooth graphs. Degenerate points are shown to have structures that follow the lines of arguments as that of the nodal sets for harmonic functions, and general PDEs.

Place, publisher, year, edition, pages
INDIANA UNIV MATH JOURNAL, 2019
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-261064 (URN)10.1512/iumj.2019.68.7711 (DOI)000484366600003 ()
Note

QC 20191001

Available from: 2019-10-01 Created: 2019-10-01 Last updated: 2019-10-01Bibliographically approved
Arakelyan, A., Barkhudaryan, R., Shahgholian, H. & Salehi, M. (2019). Numerical Treatment to a Non-local Parabolic Free Boundary Problem Arising in Financial Bubbles. Bulletin of the Iranian Mathematical Society, 45(1), 59-73
Open this publication in new window or tab >>Numerical Treatment to a Non-local Parabolic Free Boundary Problem Arising in Financial Bubbles
2019 (English)In: Bulletin of the Iranian Mathematical Society, ISSN 1018-6301, E-ISSN 1017-060X, Vol. 45, no 1, p. 59-73Article in journal (Refereed) Published
Abstract [en]

In this paper, we continue to study a non-local free boundary problem arising in financial bubbles. We focus on the parabolic counterpart of the bubble problem and suggest an iterative algorithm which consists of a sequence of parabolic obstacle problems at each step to be solved, that in turn gives the next obstacle function in the iteration. The convergence of the proposed algorithm is proved. Moreover, we consider the finite difference scheme for this algorithm and obtain its convergence. At the end of the paper, we present and discuss computational results.

Place, publisher, year, edition, pages
Springer, 2019
Keywords
Finite difference method, Viscosity solution, Free boundaries, Obstacle problem, Black-Scholes equation
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-245934 (URN)10.1007/s41980-018-0119-5 (DOI)000459196900005 ()2-s2.0-85061674630 (Scopus ID)
Note

QC 20190315

Available from: 2019-03-15 Created: 2019-03-15 Last updated: 2019-03-15Bibliographically approved
Aleksanyan, H. & Shahgholian, H. (2019). Perturbed Divisible Sandpiles and Quadrature Surfaces. Potential Analysis, 51(4), 511-540
Open this publication in new window or tab >>Perturbed Divisible Sandpiles and Quadrature Surfaces
2019 (English)In: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929X, Vol. 51, no 4, p. 511-540Article in journal (Refereed) Published
Abstract [en]

The main purpose of the present paper is to establish a link between quadrature surfaces (potential theoretic concept) and sandpile dynamics (Laplacian growth models). For this aim, we introduce a new model of Laplacian growth on the lattice DOUBLE-STRUCK CAPITAL Zd (d >= 2) which continuously deforms occupied regions of the divisible sandpile model of Levine and Peres (J. Anal. Math. 111(1), 151-219 2010), by redistributing the total mass of the system onto 1/m-sub-level sets of the odometer which is a function counting total emissions of mass from lattice vertices. In free boundary terminology this goes in parallel with singular perturbation, which is known to converge to a Bernoulli type free boundary. We prove that models, generated from a single source, have a scaling limit, if the threshold m is fixed. Moreover, this limit is a ball, and the entire mass of the system is being redistributed onto an annular ring of thickness 1/m. By compactness argument we show that when m tends to infinity sufficiently slowly with respect to the scale of the model, then in this case also there is scaling limit which is a ball, with the mass of the system being uniformly distributed onto the boundary of that ball, and hence we recover a quadrature surface in this case. Depending on the speed of decay of 1/m, the visited set of the sandpile interpolates between spherical and polygonal shapes. Finding a precise characterisation of this shape-transition phenomenon seems to be a considerable challenge, which we cannot address at this moment.

Place, publisher, year, edition, pages
SPRINGER, 2019
Keywords
Singular perturbation, Lattice growth model, Quadrature surface, Bernoulli free boundary, Boundary sandpile, Balayage, Divisible sandpile, Scaling limit
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-264863 (URN)10.1007/s11118-018-9722-6 (DOI)000496293600003 ()2-s2.0-85051720742 (Scopus ID)
Note

QC 20191217

Available from: 2019-12-17 Created: 2019-12-17 Last updated: 2020-01-03Bibliographically approved
Lee, K.-A., Park, J. & Shahgholian, H. (2019). The regularity theory for the double obstacle problem. Calculus of Variations and Partial Differential Equations, 58(3), Article ID 104.
Open this publication in new window or tab >>The regularity theory for the double obstacle problem
2019 (English)In: Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, E-ISSN 1432-0835, Vol. 58, no 3, article id 104Article in journal (Refereed) Published
Abstract [en]

In this paper, we prove a local C1-regularity of the free boundary for the (hybrid) double obstacle problem with an upper obstacle , Delta u< =f chi Omega(u)boolean AND{u<psi}+Delta psi chi Omega(u boolean AND){u=psi},u <=psi in B1, where Omega (u) = B-1\({u = 0} boolean AND {del u = 0})under a thickness assumption for u and . The novelty of the paper is the study of points where two obstacles meet, here it refers to free boundary points where =0. Our result is new, with a non-straightforward approach, as the analysis seems to require several subtle manoeuvres in finding the right conditions and methodology. A key point of difficulty lies in the classification of global solutions. This is due to the complex structure of global solutions for the double obstacle problem, and even more complex for the hybrid problem in this paper.

National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-252963 (URN)10.1007/s00526-019-1543-y (DOI)000468929600002 ()2-s2.0-85067646532 (Scopus ID)
Note

QC 20190802

Available from: 2019-08-02 Created: 2019-08-02 Last updated: 2019-08-02Bibliographically approved
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 ()2-s2.0-85062151703 (Scopus ID)
Note

QC 20180803

Available from: 2018-08-03 Created: 2018-08-03 Last updated: 2019-03-18Bibliographically approved
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Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0002-1316-7913

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