Henrik Shahgholian - profile picture on SciLag

Henrik Shahgholian

  • Analysis of PDEs
  • ArticleA double-phase problem for (p,q)-Laplacian with phase transmission


    Posted by: Henrik Shahgholian

    arXiv

    In this paper we consider the so-called double-phase problem where the phase transition takes place across the zero level "surface" of the minimizer of the functional $$ J_{p,q}(v,\Omega) = \int_\Omega \left( |\nabla v^+|^p + |\nabla v^-|^q \right) dx. $$ We prove that the minimizer exists, and is H\"older regular. From here, using an intrinsic variation, one can prove a weak formulation of the free boundary condition across the zero level surface, that formally can be represented as $$ (q-1)|\nabla u^-|^q = (p-1) |\nabla u^+|^p, \quad \hbox{on } \partial \{u > 0\}. $$ We prove that the free boundary is $C^{1,\alpha}$ a.e. with respect to the measure $\Delta_p u^+$, whose support is of $\sigma$-finite $(n-1)$-dimensional Hausdorff measure.

  • ArticleRadial symmetry for an elliptic PDE with a free boundary


    Proceeding of AMS (accepted), 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.

  • ArticleRegularity of the free boundary for a parabolic cooperative system


    Posted by: Henrik Shahgholian

    arXiv

    In this paper we study the following parabolic system \begin{equation*} \Delta {\bf u} -\partial_t {\bf u} =|{\bf u}|^{q-1}{\bf u}\,\chi_{\{ |{\bf u}|>0 \}}, \qquad {\bf u} = (u^1, \cdots , u^m) \ , \end{equation*} with free boundary $\partial \{|{\bf u} | >0\}$. For $0\leq q< 1$, we prove optimal growth rate for solutions ${\bf u} $ to the above system near free boundary points, and show that in a uniform neighbourhood of any a priori well-behaved free boundary point the free boundary is $C^{1, \alpha}$ in space directions and half-Lipschitz in the time direction.

  • ArticleThe Inhomogeneous Boundary Harnack Principle for Fully Nonlinear and p-Laplace equations


    Posted by: Henrik Shahgholian

    arXiv

    We prove a boundary Harnack principle in Lipschitz domains with small constant for fully nonlinear and $p$-Laplace type equations with a right hand side, as well as for the Laplace equation on nontangentially accessible domains under extra conditions. The approach is completely new and gives a systematic approach for proving similar results for a variety of equations and geometries.

  • ArticleLipschitz Regularity in Vectorial Linear Transmission Problems


    year of publication: 2021

    Posted by: Henrik Shahgholian

    arXiv

    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.

  • 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

    MSC 2010: 35R35

  • 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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