Henrik Shahgholian - profile picture on SciLag

Henrik Shahgholian

  • Analysis of PDEs
  • ArticleAnalysis on the steady Euler flows with stagnation points in an infinitely long nozzle


    Posted by: Henrik Shahgholian

    arXiv

    A recent prominent result asserts that steady incompressible Euler flows strictly away from stagnation in a two-dimensional infinitely long strip must be shear flows. On the other hand, flows with stagnation points, very challenging in analysis, are interesting and important phenomenon in fluids. In this paper, we not only prove the uniqueness and existence of steady flows with stagnation points, but also obtain the regularity of the boundary of stagnation set, which is a class of obstacle type free boundary.

    First, we prove a global uniqueness theorem for steady Euler system with Poiseuille flows as upstream far field state in an infinitely long strip. Due to the appearance of stagnation points, the nonlinearity of the semilinear equation for the stream function becomes non-Lipschitz. This creates a challenging analysis problem since many classical analysis methods do not apply directly. Second, the existence of steady incompressible Euler flows, tending to Poiseuille flows in the upstream, are established in an infinitely long nozzle via variational approach. A very interesting phenomenon is the regularity of the boundary of non-stagnant region, which can be regarded as an obstacle type free boundary and is proved to be globally $C^1$. Finally, the existence of stagnation region is proved as long as the nozzle is wider than the width of the nozzle at upstream where the flows tend to Poiseuille flows.

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


    Posted by: Henrik Shahgholian

    DOIarXiv

    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.

  • ArticlePointwise estimates for systems of coupled $p$-Laplacian elliptic equations


    Communications on Pure and Applied Analysis (CPAA) 22 (3), 899-921, 2023

    Posted by: Henrik Shahgholian

    DOIMSC 2010: 35J60 35B53 47J10

  • ArticleA transmission problem with ($p,q$)-Laplacian


    Communications in Partial Differential Equations 48 (2), 315-349, 2023

    Posted by: Henrik Shahgholian

    DOIMSC 2010: 35A15

    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.

  • ArticleAlmost minimizers for a sublinear system with free boundary


    Calculus of Variations and Partial Differential Equations 62 (5), Paper No. 149, 43, 2023

    Posted by: Henrik Shahgholian

    DOIMSC 2010: 35R35 35J60

  • ArticleOn global solutions of the obstacle problem


    Duke Mathematical Journal 172 (11), 2149-2193, 2023

    Posted by: Henrik Shahgholian

    DOIMSC 2010: 35R35 35J05

  • ArticleThe structure of the regular part of the free boundary close to singularities in the obstacle problem


    Journal of Differential Equations 377, 873-887, 2023

    Posted by: Henrik Shahgholian

    DOIMSC 2010: 35J25

  • Article"Women History Month”: Nina Nikolaevna Uraltseva


    Notices of the American Mathematical Society 69 (3), 385–395, 2022

    Posted by: Henrik Shahgholian

    arXiv

    Nina Uraltseva has made lasting contributions to mathematics with her pioneering work in various directions in analysis and PDEs and the development of elegant and sophisticated analytical techniques. She is most renowned for her early work on linear and quasilinear equations of elliptic and parabolic type in collaboration with Olga Ladyzhenskaya, which is the category of classics, but her contributions to the other areas such as degenerate and geometric equations, variational inequalities, and free boundaries are equally deep and significant. In this article, we give an overview of Nina Uraltseva's work with some details on selected results.

  • ArticleInhomogeneous global minimizers to the one-phase free boundary problem


    Communications in Partial Differential Equations (accepted) 47 (6), 1193--1216, 2022

    Posted by: Henrik Shahgholian

    arXivfulltext

    Given a global 1-homogeneous minimizer $U_0$ to the Alt-Caffarelli energy functional, with $sing(F(U_0)) = \{0\}$, we provide a foliation of the half-space ${\mathbb R}^{n} \times [0,+\infty)$ with dilations of graphs of global minimizers $\underline U \leq U_0 \leq \bar U$ with analytic free boundaries at distance 1 from the origin.

  • ArticleRegularity of the free boundary for a parabolic cooperative system


    Calculus of Variations and Partial Differential Equations 61 (4), 2022

    Posted by: Henrik Shahgholian

    arXivfulltext

    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.

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