Related papers: H\"older regularity for nonlocal double phase equa…

We prove the local boundedness and the local H\"older continuity of weak solutions to nonlocal equations with variable orders and exponents under sharp assumptions.

Let $D$ be an bounded region in ${\bf R}^n$. The regularity of solutions of a family of quasilinear elliptic partial differential equations is studied, one example being $\Delta_nu=Vu^{n-1}$. The coefficients are assumed to be in the space…

We prove H\"older continuous regularity of bounded, uniformly continuous, viscosity solutions of degenerate fully nonlinear equations defined in all of $\mathbb{R}^n$ space. In particular the result applies also to some operators in Carnot…

In this work, we present a systematic approach to investigate the existence, multiplicity, and local gradient regularity of solutions for nonlocal quasilinear equations with local gradient degeneracy. Our method involves an interactive…

We investigate a class of degenerate fully nonlinear nonlocal elliptic equations with Hamiltonian terms. By precisely characterizing the interaction between the degeneracy law of equations and the growth behavior of the Hamiltonian terms,…

In this paper, we obtain local H\"older regularity for bounded, weak solutions to the anisotropic $p$-Laplace equation whose prototype structure is given by $$ \sum_{i=1}^N (|u_{x_i}|^{p_i-2}u_{x_i})_{x_i}=0,$$ where $1 < p_1 \leq p_2 \leq…

We prove the H\"{o}lder continuity of sign-changing solutions to the equation of the type $$\frac{\partial}{\partial t}\big(|u|^{q-1} u\big)- div\Big(|D u|^{p-2}\,D u\Big)=0,$$ where numbers $p$, $q$ satisfy the conditions $$0<q<p-1\quad…

We prove existence, uniqueness, and regularity of viscosity solutions to the stationary and evolution obstacle problems defined by a class of nonlocal operators that are not stable-like and may have supercritical drift. We give sufficient…

We prove optimal regularity results in $L_p$-based function spaces in space and time for a large class of linear parabolic equations with a nonlocal elliptic operator in bounded domains with limited smoothness. Here the nonlocal operator is…

We prove that bounded solutions to degenerate parabolic double-phase problem modelled upon \[u_t-\dv(|\na u|^{p-2}\na u+a(x,t)|\na u|^{q-2}\na u)=-\dv(|F|^{p-2}F+a(x,t)|F|^{q-2}F)\,, \] where a nonnegative weight $a$ is $\alpha$-H\"older…

For $2a$-order strongly elliptic operators $P$ generalizing $(-\Delta )^a$, $0<a<1$, the treatment of the homogeneous Dirichlet problem on a bounded open set $\Omega \subset R^n$ by pseudodifferential methods, has been extended in a recent…

We prove a quantitative inhomogeneous Hopf-Oleinik lemma for viscosity solutions of $$|\nabla u|^{\alpha}F(D^{2}u)=f $$ and, more generally, for viscosity supersolutions of $|\nabla u|^{\alpha}\,{M}^-_{\lambda,\Lambda}(D^{2}u)\le f$. The…

We study well-posedness of degenerate mixed-type parabolic-hyperbolic equations $$ \partial_tu+\text{div}\big(f(u)\big)=\mathcal{L}[b(u)] $$ on bounded domains with general Dirichlet boundary/exterior conditions. The nonlocal diffusion…

We prove optimal boundary $C^{1,\alpha}$ regularity for viscosity solutions of degenerate fully nonlinear uniformly elliptic equations with oblique boundary conditions and Hamiltonian terms of the form \[ \begin{cases} |Du|^{\gamma}F(D^2 u)…

We introduce the concept of $C^{m,\alpha}$-nonlocal operators, extending the notion of second order elliptic operator in divergence form with $C^{m,\alpha}$-coefficients. We then derive the nonlocal analogue of the key existing results for…

This paper is a continuation of our previous work [21], where we have established that, for the second-order degenerate hyperbolic equation (\p_t^2-t^m\Delta_x)u=f(t,x,u), locally bounded, piecewise smooth solutions u(t,x) exist when the…

We consider the following mixed local and non-local critical elliptic equation: \begin{equation*}\label{0.1} \left\{ \begin{array}{lll} -\Delta u+(-\Delta)^su=\lambda h u^{p}+u^{2^*-1}, &\text{in}\,\, \mathbb{R}^n, u>0, &\text {in} \,\,…

In this paper we study the local regularity properties of weak solutions to a special class of anisotropic doubly nonlinear parabolic operators, whose prototype is the anisotropic Trudinger's equation $$ u_t- \sum\limits_{i=1}^N…

In this article we prove for the first time the $C^s$ boundary regularity for solutions to nonlocal elliptic equations with H\"older continuous coefficients in divergence form in $C^{1,\alpha}$ domains. So far, it was only known that…

We investigate the regularity of the viscosity solutions to a class of degenerate/singular fully nonlinear elliptic equations with Hamiltonian terms. To overcome the difficulty caused by the simultaneous presence of the general…