Henrik Shahgholian

Abstract: 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, seems likely to adapt to more general domains, as well as other types of operators.

We consider operators of the form $$ \mathcal{L} u = (a^{ij}u_i)_j + b^i u_i + cu , $$ with the following ellipticity conditions $$ \Lambda^{-1} |\xi|^2 \leq \langle a^{ij}(x) \xi, \xi \rangle \leq \Lambda |\xi|^2, $$ for some $\Lambda >0$ and for all nonzero $\xi \in \R^n$. Furthermore, $a^{ij}(x)$ is a real $n \times n$ matrix. For the lower order terms we assume $$|c(x)|, \sum |b^i(x)| \leq \Lambda-1$$ and that $c(x) \leq 0$. We say $u \in \mathcal{S_{\mathcal{L}}}(\Omega_{L,R})$ if $$ \begin{aligned} \mathcal{L} u(x) &=0 \text{ in } \Omega_{L,R} \cap B_R ,\\ u(x)&=0 \text{ on } \Omega_{L,R}^c\cap B_R, \end{aligned} $$ and that $u \in \mathcal{S_{\mathcal{L}}}(\Omega_{L,R},d^{\gamma})$ if $$ \begin{aligned} |\mathcal{L} u(x)| &\leq (\text{dist}(x,\partial \Omega_{L,R} \cap B_R))^{\gamma} \ \text{ in } \Omega_{L,R} , \\ u(x)&=0 \ \text{ on } \Omega_{L,R}^c\cap B_R. \end{aligned} $$ To apply the Hölder continuity estimates for elliptic operators we will require that $\gamma>-2/n $. Since the boundary is Lipschitz, this will ensure the correct integrability assumptions for the right hand side.

Theorem:

Let $0 \in \partial \Omega_L$ with $L< M$, and fix $x^0 \in \Omega_L$. Assume further that $B_1 \cap \{x_n > 1/4\} \subseteq \Omega_L$. Assume $u,v \geq 0$ and $u,v \in \mathcal{S}(\Omega_{L},d^{\gamma})$ with $\mathcal L u, \mathcal L v \leq 0$ and $u(x^0)=v(x^0)=1$, and also assume that $2-\alpha+\gamma>0$, with $\gamma>-2/n$ and $\alpha$ be such that $$ \sup_{B_{r}(x)} u \geq c_1 u(e_n/2) r^{\alpha}. $$

Then there exists a uniform constant $C>0$ (depending only on dimension $n$, Lipschitz constant $M$, $M-L$, and dist$(x^0,\partial \Omega_L)$) such that \begin{equation} \label{e:ine3} C^{-1} v(x) \leq u(x) \leq Cv(x) \end{equation} for all $x \in B_{1/2}$.

This paper studies the fully nonlinear free boundary problem $$ F(D^2u)=1 \ \text{a.e. in }B_1 \cap \Omega \qquad \hbox{and }\qquad |D^2 u| \leq K , \ \text{a.e. in }B_1\setminus\Omega, $$ where $K>0$, and $\Omega$ is an unknown open set. The main result is the optimal regularity for solutions to this problem: namely, we prove that $W^{2,n}$ solutions are locally $C^{1,1}$ inside $B_1$. Under the extra condition that $ {\Omega \supset \{D{u} \neq 0 \}}$, and a uniform thickness assumption on the coincidence set $\{Du=0\}$. It is also shown a local regularity for the free boundary $\partial \Omega \cap B_1$.

In this paper we initiate the study of maps minimising the energy $$ \int_{D} (|\nabla {\mathbf u}|^2+2| {\mathbf u}|)\ dx. $$ which, due to Lipschitz character of the integrand, gives rise to the singular Euler equations $$ \Delta {\mathbf u}=\frac{ {\mathbf u}}{| {\mathbf u}|}\chi_{\left\lbrace | {\mathbf u}|>0\right\rbrace}, \qquad {\mathbf u} = (u_1, \cdots, u_m) \ . $$ Our primary goal in this paper is to set up a road map for future developments of the theory related to such energy minimising maps. Our results here concern regularity of the solution as well as that of the free boundary. They are achieved by using monotonicity formulas and epiperimetric inequalities, in combination with geometric analysis.

We study the local behavior of the nodal sets of the solutions to elliptic quasilinear equations with nonlinear conductivity part, $\hbox{div}(A_s (x, u)u) = \hbox{div} { \mathbf f (x)}$, where $A_s (x, u)$ has " broken " derivatives of order $s \geq 0$, 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 $\mathbf f (x)$ is assumed to be $C^\alpha$ in case $s = 0$, and $C^{ 1,\alpha}$ (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 exactly, 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.

In this paper we consider a quasilinear elliptic PDE, $\hbox{div} (A(x,u) \nabla u) =0$, where the underlying physical problem gives rise to a jump for the conductivity $A(x,u)$, across a level surface for $u$. Our analysis concerns Lipschitz regularity for the solution $u$, and the regularity of the level surfaces, where $A(x,u)$ has a jump and the solution $u$ does not degenerate. In proving Lipschitz regularity of solutions, we introduce a new and unexpected type of ACF-monotonicity formula with two different operators, that might be of independent interest, and surely can be applied in other related situations. The proof of the monotonicity formula is done through careful computations, and (as a byproduct) a slight generalization to a specific type of variable matrix-valued conductivity is presented.