Henrik Shahgholian - profile picture on SciLag

Henrik Shahgholian

  • Analysis of PDEs
  • ArticleLipschitz Regularity in Vectorial Linear Transmission Problems

    Nonlinear Analysis, 2022

    Posted by: Henrik Shahgholian


    We consider vector-valued solutions to a linear transmission problem, and we prove that Lipschitz-regularity on one phase is transmitted to the next phase. More exactly, given a solution $u:B_1\subset {\mathbb R}^n \to {\mathbb R}^m$ to the elliptic system \begin{equation*} \hbox{div} ((A + (B-A)\chi_D )\nabla u) = 0 \quad \text{in }B_1, \end{equation*} where $A$ and $B$ are Dini continuous, uniformly elliptic matrices, we prove that if $\nabla u \in L^{\infty} (D)$ then $u$ is Lipschitz in $B_{1/2}$. A similar result is also derived for the parabolic counterpart of this problem.

  • ArticleOptimal regularity for a two-phase obstacle-like problem with logarithmic singularity

    Communications in Partial Differential Equations 46 (10), 1831-1850, 2021

    Posted by: Henrik Shahgholian

    DOIMSC 2010: 35R35 35J61

  • ArticleA free boundary problem for an elliptic system

    Journal of Differential Equations 284, 126-155, 2021

    Posted by: Henrik Shahgholian

    DOIMSC 2010: 35J56

  • ArticleFree boundary methods and non-scattering phenomena

    Research in the Mathematical Sciences 8 (4), 2021

    Posted by: Henrik Shahgholian


    We study a question arising in inverse scattering theory: given a penetrable obstacle, does there exist an incident wave that does not scatter? We show that every penetrable obstacle with real-analytic boundary admits such an incident wave. At zero frequency, we use quadrature domains to show that there are also obstacles with inward cusps having this property. In the converse direction, under a nonvanishing condition for the incident wave, we show that there is a dichotomy for boundary points of any penetrable obstacle having this property: either the boundary is regular, or the complement of the obstacle has to be very thin near the point. These facts are proved by invoking results from the theory of free boundary problems.

  • ArticleRadial symmetry for an elliptic PDE with a free boundary

    Proceedings of the American Mathematical Society. Series B (accepted) 8, 311-319, 2021

    Posted by: Henrik Shahgholian


    In this paper we prove symmetry for solutions to the semi-linear elliptic equation $$ \Delta u = f(u) \quad \hbox{ in } B_1, \qquad 0 \leq u < M, \quad \hbox{ in } B_1, \qquad u = M, \quad \hbox{ on } \partial B_1, $$ where $M>0$ is a constant, and $B_1$ is the unit ball. Under certain assumptions on the r.h.s. $f (u)$, the $C^1$-regularity of the free boundary $\partial \{u>0\}$ and a second order asymptotic expansion for $u$ at free boundary points, we derive the spherical symmetry of solutions. A key tool, in addition to the classical moving plane technique, is a boundary Harnack principle (with r.h.s.) that replaces Serrin's celebrated boundary point lemma, which is not available in our case due to lack of $C^2$-regularity of solutions.

  • ArticleMonotonicity formulas for coupled elliptic gradient systems with applications

    Advances in Nonlinear Analysis 9 (1), 479-495, 2020

    Posted by: Henrik Shahgholian

    DOIMSC 2010: 35R11 35B35 35B45 35J47 35J50

  • ArticleIsolated singularities for semilinear elliptic systems with power-law nonlinearity

    Analysis & PDE 13 (3), 701-739, 2020

    Posted by: Henrik Shahgholian

    DOIMSC 2010: 35J91 35B40 35B65 35C20 35J75

  • ArticleRemarks on the decay/growth rate of solutions to elliptic free boundary problems of obstacle type

    Mathematics in Engineering 2 (4), 698-708, 2020

    Posted by: Henrik Shahgholian

    DOIMSC 2010: 35R35 35J75 35J86

  • ArticleRemarks on the convexity of free boundaries (scalar and system cases)

    Rossi\u\iskaya Akademiya Nauk. Algebra i Analiz 32 (4), 146-166, 2020

    Posted by: Henrik Shahgholian

    fulltextMSC 2010: 35R35

    We discuss convexity for several free boundary value problems in exterior domains that are generally formulated as $$ \Delta u = f(u) \quad \text{in } \Omega \setminus D, \qquad |\nabla u | = g \quad \text{on } \partial \Omega , \qquad u\geq 0 \quad \text{in } \Bbb R^n $$ where $u$ is assumed to be continuous in $ \Bbb R^n$, $ \Omega = \{u > 0\}$ (is unknown), $u=1$ on $\partial D$, and $D$ is a bounded domain in $\Bbb R^n$ ($n\geq 2$). Here $g= g(x)$ is a given smooth function which is either strictly positive (Bernoulli-type) or identically zero (obstacle type). Properties for $f$ will be spelled out in exact terms in the text.

    We make the following general assumptions on $f$ (see [Fr-Ph] for similar type of problems) $$ \begin{cases} f(t) > 0 \qquad & \text{for} \,\,t > 0 , \\ f(t)=0 \qquad & \text{for} \,\, t\leq 0 \\ f(t)= b t^{\alpha} + o(t^{\alpha}) \qquad & \text{for}\,\, t < t_1, \,\, \text{some}\,\, t_1 >0 \,\,\text{and} \,\, -1< \alpha< 1,\\ \end{cases} $$ where $ b \geq 0$. We also assume $$ \hbox{ $f$ is either left- or right-continuous and its discontinuities are isolated}. $$


    Let $D$ be a bounded convex domain in $\Bbb R^n$ ($n \geq 2$), and $f$ satisfy the above properties. Let $g(x)$ be a $C^\beta$-function in the entire space satisfying $\max (b, g(x)) \geq b_0$ for some $b_0$, and $g(x) $ be either identically zero or a strict positive. When $g > 0$ we assume $1/g$ to be concave function.

    Then there exists a non-negative function $u$ with $\Omega := \{ u >0\}$ solving the free boundary problem $$ \begin{cases} \Delta u = f(u) \qquad & \text{in } \Omega \setminus D , \\ u = 1 \qquad& \text{on } \partial D ,\\ |\nabla u |= g \qquad& \text{on } \partial \Omega , \end{cases} $$ with convex super-level sets i.e. $\{u > l \}$ is convex, for all $0 \leq l \leq 1$.

    If in addition there is a point $z \in D$, such that $$ tg(t(x-z) + z) \ \hbox{is non-decreasing in } t, \ \hbox{ for all } x \in \Bbb R^n, $$ then $u$ is the unique solution.

  • ArticleNumerical treatment to a non-local parabolic free boundary problem arising in financial bubbles

    Bulletin of the Iranian Mathematical Society 45 (1), 59-73, 2019

    Posted by: Henrik Shahgholian

    DOIMSC 2010: 65M06 35D40 35R35 65M12 91G80

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