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Henrik Shahgholian

Publications and source records attributed to Henrik Shahgholian.

At least 37 records · Page 2Linked to original sources
arXiv2024

On scattering behavior of corner domains with anisotropic inhomogeneities

This paper investigates the possible scattering and non-scattering behavior of an anisotropic and inhomogeneous Lipschitz medium at a fixed wave number and with a single incident field. We connect the anisotropic non-scattering problem to a Bernoulli type free boundary problem. By invoking methods from the theory of free boundaries, we show that an anisotropic medium with Lipschitz but not $C^{1,α}$ boundary scatters every incident wave that satisfies a non-degeneracy condition.

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arXiv2024

Convexity for a parabolic fully nonlinear free boundary problem with singular term

In this paper, we study a parabolic free boundary problem in an exterior domain $$\begin{cases} F(D^2u)-\partial_tu=u^aχ_{\{u>0\}}&\text{in }(\mathbb R^n\setminus K)\times(0,\infty),\\ u=u_0&\text{on }\{t=0\},\\ |\nabla u|=u=0&\text{on }\partialΩ\cap(\mathbb R^n\times(0,\infty)),\\ u=1&\text{in }K\times[0,\infty).\end{cases}$$ Here, $a$ belongs to the interval $(-1,0)$, $K$ is a (given) convex compact set in $\mathbb R^n$, $Ω=\{u>0\}\supset K\times(0,\infty)$ is an unknown set, and $F$ denotes a fully nonlinear operator. Assuming a suitable condition on the initial value $u_0$, we prove the existence of a nonnegative quasiconcave solution to the aforementioned problem, which exhibits monotone non-decreasing behavior over time.

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arXiv2023

Analysis on the steady Euler flows with stagnation points in an infinitely long nozzle

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.

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arXiv2023

Almost minimizers to a transmission problem for $(p,q)$-Laplacian

This paper concerns almost minimizers of the functional $$ J(v,Ω) = \int_Ω\left( |D v^+|^p + |D v^-|^q \right) dx, $$ where $1<p \neq q< \infty$ and $Ω$ is a bounded domain of $\mathbb{R}^n$, $n\geq 1$. We prove the universal Hölder regularity of local $(1+ε)$-minimizers, when $ε$ is universally small. Moreover, we prove almost Lipschitz regularity of the local $(1+ε)$-minimizers, when $|p-q|\ll 1$ and $ε\ll 1$.

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arXiv2023

Lipschitz regularity of a weakly coupled vectorial almost-minimizers for the $p$-Laplacian

For a given constant $λ> 0$ and a bounded Lipschitz domain $D \subset \mathbb{R}^n$ ($n \geq 2$), we establish that almost-minimizers of the functional $$ J(\mathbf{v}; D) = \int_D \sum_{i=1}^{m} \left|\nabla v_i(x) \right|^p+ λχ_{\{\left|\mathbf{v} \right|>0\}} (x) \, dx, \qquad 1<p<\infty, $$ where $\mathbf{v} = (v_1, \cdots, v_m)$, and $m \in \mathbb{N}$, exhibit optimal Lipschitz continuity in compact sets of $D$. Furthermore, assuming $p \geq 2$ and employing a distinctly different methodology, we tackle the issue of boundary Lipschitz regularity for $v$. This approach simultaneously yields alternative proof for the optimal local Lipschitz regularity for the interior case.

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arXiv2023

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

We prove the -- to the best knowledge of the authors -- first result on the fine asymptotic behavior of the regular part of the free boundary of the obstacle problem close to singularities. The result is motivated by our recent partial answer to a long standing conjecture and the first partial classification of global solutions of the obstacle problem with unbounded coincidence sets in higher dimensions.

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arXiv2023

On positivity sets for Helmholtz solutions

We address the question of finding global solutions of the Helmholtz equation that are positive in a given set. This question arises in inverse scattering for penetrable obstacles. In particular, we show that there are solutions that are positive on the boundary of a bounded Lipschitz domain.

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arXiv2023

Constraint maps with free boundaries: the obstacle case

This paper revives a four-decade-old problem concerning regularity theory for (continuous) constraint maps with free boundaries. Dividing the map into two parts, the distance part and the projected image to the constraint, one can prove various properties for each component. As already pointed out in the literature, the distance part falls under the classical obstacle problem, which is well-studied by classical methods. A perplexing issue, untouched in the literature, is the properties of the projected image and its higher regularity, which we show to be at most of class $C^{2,1}$. In arbitrary dimensions, we prove that the image map is globally of class $W^{3,BMO}$, and locally of class $C^{2,1}$ around the regular part of the free boundary. The issue becomes more delicate around singular points, and we resolve it in two dimensions. In the appendix, we extend some of our results to what we call leaky maps.

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arXiv2022

Convexity for free boundaries with singular term (nonlinear elliptic case)

We consider a free boundary problem in an exterior domain \begin{cases}\begin{array}{cc} Lu=g(u) & \text{in }Ω\setminus K, \\ u=1 & \text{on }\partial K,\\ |\nabla u|=0 &\text{on }\partial Ω, \end{array}\end{cases} where $K$ is a (given) convex and compact set in $\mathbb{R}^n$ ($n\ge2$), $Ω=\{u>0\}\supset K$ is an unknown set, and $L$ is either a fully nonlinear or the $p$-Laplace operator. Under suitable assumptions on $K$ and $g$, we prove the existence of a nonnegative quasi-concave solution to the above problem. We also consider the cases when the set $K$ is contained in $\{x_n=0\}$, and obtain similar results.

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arXiv2022

A transmission problem for $(p,q)$-Laplacian

In this paper, we consider a double-phase problem characterised by a transmission that takes place across the zero level "surface" of the minimiser of the functional $$ J(v,Ω) = \int_Ω\left( |D v^+|^p + |D v^-|^q \right) dx. $$ We prove that a minimiser exists, and is Hölder continuous, whence using an intrinsic variation we prove a weak formulation of the free boundary condition across the zero level surface, formally represented by $$ (q-1)|D u^-|^q = (p-1) |D u^+|^p, \quad \hbox{on } \partial \{u > 0\}. $$ We show that the free boundary is $C^{1,α}$ a.e. with respect to the measure $Δ_p u^+$, whose support is of $σ$-finite $(n-1)$-dimensional Hausdorff measure.

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arXiv2022

On global solutions of the obstacle problem

Assuming a lower bound on the dimension, we prove a long standing conjecture concerning the classification of global solutions of the obstacle problem with unbounded coincidence sets.

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arXiv2022

Almost minimizers for a sublinear system with free boundary

We study vector-valued almost minimizers of the energy functional $$\int_D\left(|\nabla\mathbf{u}|^2+\frac2{1+q}\left(λ_+(x)|\mathbf{u}^+|^{q+1}+λ_-(x)|\mathbf{u}^-|^{q+1}\right)\right)dx,\quad0 0$, we take the epiperimetric inequality approach and prove the regularity for both almost minimizers and the set of "regular" free boundary points.

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arXiv2022

Symmetry for a fully nonlinear free boundary problem with highly singular term

In this paper we prove radial symmetry for solutions to a free boundary problem with a singular right hand side, in both elliptic and parabolic regime. More exactly, in the unit ball $B_1$ we consider a solution to the fully nonlinear elliptic problem $$ \begin{cases} F(D^2u)=f(u)&\text{in }B_1 \cap \{u >0 \},\\ u=M&\text{on }\partial B_1,\\ 0\le u 0\}$, we cannot apply the well-known Serrin-type boundary point lemma. We circumvent this by an exact assumption on a first order expansion and the decay on the second order, along with an ad-hoc comparison principle. We treat equally the parabolic case of the problem, and state a corresponding result.

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arXiv2022

Nina Nikolaevna Uraltseva

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.

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