Nonlocal elliptic equations in bounded domains: a survey

In this paper we survey some results on the Dirichlet problem $$\left\{ \begin{array}{rcll} L u &=&f&\textrm{in }\Omega \\ u&=&g&\textrm{in }\mathbb R^n\backslash\Omega \end{array}\right.$$ for nonlocal operators of the form $$Lu(x)=\textrm{PV}\int_{\mathbb R^n}\bigl\{u(x)-u(x+y)\bigr\}K(y)dy.$$ We start from the very basics, proving existence of solutions, maximum principles, and constructing some useful barriers. Then, we focus on the regularity properties of solutions, both in the interior and on the boundary of the domain. In order to include some natural operators $L$ in the regularity theory, we do not assume any regularity on the kernels. This leads to some interesting features that are purely nonlocal, in the sense that have no analogue for local equations. We hope that this survey will be useful for both novel and more experienced researchers in the field

Boundary regularity, Pohozaev identities, and nonexistence results

In this expository paper we survey some recent results on Dirichlet problems of the form $Lu=f(x,u)$ in $\Omega$, $u\equiv0$ in $\mathbb R^n\backslash\Omega$. We first discuss in detail the boundary regularity of solutions, stating the main known results of Grubb and of the author and Serra. We also give a simplified proof of one of such results, focusing on the main ideas and on the blow-up techniques that we developed in \cite{RS-K,RS-stable}. After this, we present the Pohozaev identities established in \cite{RS-Poh,RSV,Grubb-Poh} and give a sketch of their proofs, which use strongly the fine boundary regularity results discussed previously. Finally, we show how these Pohozaev identities can be used to deduce nonexistence of solutions or unique continuation properties. The operators $L$ under consideration are integro-differential operator of order $2s$, $s\in(0,1)$, the model case being the fractional Laplacian $L=(-\Delta)^s$.Comment: Survey article. To appear as a chapter in "Recent Developments in the Nonlocal Theory" by De Gruyte

Regularity of stable solutions up to dimension 7 in domains of double revolution

We consider the class of semi-stable positive solutions to semilinear equations $-\Delta u=f(u)$ in a bounded domain $\Omega\subset\mathbb R^n$ of double revolution, that is, a domain invariant under rotations of the first $m$ variables and of the last $n-m$ variables. We assume $2\leq m\leq n-2$. When the domain is convex, we establish a priori $L^p$ and $H^1_0$ bounds for each dimension $n$, with $p=\infty$ when $n\leq7$. These estimates lead to the boundedness of the extremal solution of $-\Delta u=\lambda f(u)$ in every convex domain of double revolution when $n\leq7$. The boundedness of extremal solutions is known when $n\leq3$ for any domain $\Omega$, in dimension $n=4$ when the domain is convex, and in dimensions $5\leq n\leq9$ in the radial case. Except for the radial case, our result is the first partial answer valid for all nonlinearities $f$ in dimensions $5\leq n\leq 9$

Sobolev and isoperimetric inequalities with monomial weights

We consider the monomial weight $|x_1|^{A_1}...|x_n|^{A_n}$ in $\mathbb R^n$, where $A_i\geq0$ is a real number for each $i=1,...,n$, and establish Sobolev, isoperimetric, Morrey, and Trudinger inequalities involving this weight. They are the analogue of the classical ones with the Lebesgue measure $dx$ replaced by $|x_1|^{A_1}...|x_n|^{A_n}dx$, and they contain the best or critical exponent (which depends on $A_1$, ..., $A_n$). More importantly, for the Sobolev and isoperimetric inequalities, we obtain the best constant and extremal functions. When $A_i$ are nonnegative \textit{integers}, these inequalities are exactly the classical ones in the Euclidean space $\mathbb R^D$ (with no weight) when written for axially symmetric functions and domains in $\mathbb R^D=\mathbb R^{A_1+1}\times...\times\mathbb R^{A_n+1}$.Comment: The proof of Theorem 1.6 in the previous version of this paper was not correct. Indeed, Lemma 5.1 in that version was not true as stated therein. We thank Georgios Psaradakis for pointing this to u

Boundary regularity for fully nonlinear integro-differential equations

We study fine boundary regularity properties of solutions to fully nonlinear elliptic integro-differential equations of order $2s$, with $s\in(0,1)$. We consider the class of nonlocal operators $\mathcal L_*\subset \mathcal L_0$, which consists of infinitesimal generators of stable L\'evy processes belonging to the class $\mathcal L_0$ of Caffarelli-Silvestre. For fully nonlinear operators $I$ elliptic with respect to $\mathcal L_*$, we prove that solutions to $I u=f$ in $\Omega$, $u=0$ in $\mathbb R^n\setminus\Omega$, satisfy $u/d^s\in C^{s+\gamma}(\overline\Omega)$, where $d$ is the distance to $\partial\Omega$ and $f\in C^\gamma$. We expect the class $\mathcal L_*$ to be the largest scale invariant subclass of $\mathcal L_0$ for which this result is true. In this direction, we show that the class $\mathcal L_0$ is too large for all solutions to behave like $d^s$. The constants in all the estimates in this paper remain bounded as the order of the equation approaches 2. Thus, in the limit $s\uparrow1$ we recover the celebrated boundary regularity result due to Krylov for fully nonlinear elliptic equations.Comment: To appear in Duke Math.

Entire solutions to semilinear nonlocal equations in \RR^2

We consider entire solutions to $L u= f(u)$ in \RR^2, where $L$ is a general nonlocal operator with kernel $K(y)$. Under certain natural assumtions on the operator $L$, we show that any stable solution is a 1D solution. In particular, our result applies to any solution $u$ which is monotone in one direction. Compared to other proofs of the De Giorgi type results on nonlocal equations, our method is the first successfull attempt to use the Liouville theorem approach to get flatness of the level sets