Related papers: \epsilon-regularity for systems involving non-loca…
Local boundedness and Harnack inequalities are studied for solutions to parabolic and elliptic integro-differential equations whose governing nonlocal operators are associated with nonsymmetric forms. We present two independent proofs, one…
We prove higher regularity for nonlinear nonlocal equations with possibly discontinuous coefficients of VMO-type in fractional Sobolev spaces. While for corresponding local elliptic equations with VMO coefficients it is only possible to…
We survey recent regularity results for parabolic equations involving nonlocal operators like the fractional Laplacian. We extend the results of Felsinger-Kassmann (2013) and obtain regularity estimates for nonlocal operators with kernels…
Despite significant recent advances in the regularity theory for obstacle problems with integro-differential operators, some fundamental questions remained open. On the one hand, there was a lack of understanding of parabolic problems with…
The primary purpose of this paper is to study the Wiener-type regularity criteria for non-linear equations driven by integro-differential operators, whose model is the fractional $p-$Laplace equation. In doing so, with the help of tools…
In the elliptic theory for $p$-Laplacian-like problems, the H\"{o}lder continuity of solutions has been proven for problems arising as Euler--Lagrange equations of a convex potential with $p$-growth that additionally satisfies the splitting…
In this paper we consider minimizers for nonlocal energy functionals generalizing elastic energies that are connected with the theory of peridynamics \cite{Silling2000} or nonlocal diffusion models \cite{Rossi}. We derive nonlocal versions…
We study the regularity for solutions of fully nonlinear integro differential equations with respect to nonsymmetric kernels. More precisely, we assume that our operator is elliptic with respect to a family of integro differential linear…
Local asymptotic stabilizability is a topic of great theoretical interest and practical importance. Broadly, if a system $\dot{x} = f(x,u)$ is locally asymptotically stabilizable, we are guaranteed a feedback controller $u(x)$ that forces…
In this paper we prove unique continuation principles for some systems of elliptic partial differential equations satisfying a suitable superlinearity condition. As an application, we obtain nonexistence of nontrivial (not necessarily…
We propose a geometric setup to study analytic aspects of a variant of the super symmetric two-dimensional nonlinear sigma model. This functional extends the functional of Dirac-harmonic maps by gravitino fields. The system of…
We introduce a new class of fully nonlinear integro-differential operators with possible nonsymmetric kernels, which includes the ones that arise from stochastic control problems with purely jump L\`evy processes. If the index of the…
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…
We prove results on solvability of nonlinear elliptic partial differential systems of principle type of second order. They are consequences of existence of non-radial solutions for nonlinear partial differential systems of Poisson type. As…
We prove a $C^{1,\alpha}$ interior regularity theorem for fully nonlinear uniformly elliptic integro-differential equations without assuming any regularity of the kernel. We then give some applications to linear theory and higher regularity…
In this survey we prove H\"older regularity results for viscosity solutions of fully nonlinear nonlocal uniformly elliptic second order differential equations with local gradient terms. This extends the nonlocal counterpart of the work of…
We give new estimates for a critical elliptic system introduced by Rivi\`ere-Struwe in \cite{riviere_struwe} (see also the work of Rupflin \cite{rupflin} and Schikorra \cite{schikorra_frames}), which generalises PDE solved by harmonic (and…
Two main results will be presented in our paper. First, we will prove the regularity of solutions to axially symmetric Navier-Stokes equations under a $log$ supercritical assumption on the horizontally radial component $u^r$ and vertical…
We use the formalism of the R{\'e}nyi entropies to establish the symmetry range of extremal functions in a family of subcriti-cal Caffarelli-Kohn-Nirenberg inequalities. By extremal functions we mean functions which realize the equality…
We obtain sharp local $C^{1,\alpha}$ regularity of solutions for singular obstacle problems, Euler-Lagrange equation of which is given by $$ \Delta_p u=\gamma(u-\varphi)^{\gamma-1}\,\text{ in }\,\{u>\varphi\}, $$ for $0<\gamma<1$ and…