Related papers: Regularity theory for Second order Integro-PDEs
In this paper we show how to derive regularity for the solution of Kolmogorov PIDEs driven by a vector field which is a second order integro differential operator of affine type. These results are valuable in applications, in particular for…
We develop an optimal regularity theory for parabolic partial differential equations in weighted mixed norm Sobolev-Zygmund spaces. The results extend the classical Schauder estimates to coefficients that are merely measurable in time and…
The regularity and characterization of solutions to degenerate, quasilinear SPDE is studied. Our results are two-fold: First, we prove regularity results for solutions to certain degenerate, quasilinear SPDE driven by Lipschitz continuous…
We prove H\"older estimates for viscosity solutions of a class of possibly degenerate and singular equations modelled by the fractional $p$-Laplace equation $$ \text{PV}…
In the first part of planned series of papers the formal general solutions to selection of 80 examples of different types of second order nonlinear PDEs in two independent variables with constant parameters are given. The main goal here is…
We prove new velocity averaging results for second-order multidimensional equations of the general form, $\op(\nabla_x,v)f(x,v)=g(x,v)$ where $\op(\nabla_x,v):=\bba(v)\cdot\nabla_x-\nabla_x^\top\cdot\bbb(v)\nabla_x$. These results quantify…
We establish the local H\"older regularity of the spatial gradient of bounded weak solutions $u\colon E_T\to\R^k$ to the non-linear system of parabolic type \begin{equation*} \partial_tu-\Div\Big(…
In this paper, we propose an approximation method to study the regularity of solutions to the Isaacs equation. This class of problems plays a paramount role in the regularity theory for fully nonlinear elliptic equations. First, it is a…
In this paper, we show that the minimal solution of a backward stochastic differential equation gives a probabilistic representation of the minimal viscosity solution of an integro-partial differential equation both with a singular terminal…
Existence, uniqueness, and regularity of a strong solution are obtained for stochastic PDEs with a colored noise $F$ and its super-linear diffusion coefficient: $$ du=(a^{ij}u_{x^ix^j}+b^iu_{x^i}+cu)dt+\xi|u|^{1+\lambda}dF, \quad…
We classify regularity for Lagrangian mean curvature type equations, which include the potential equation for prescribed Lagrangian mean curvature and those for Lagrangian mean curvature flow self-shrinkers and expanders, translating…
In the analysis of PDEs, regularity of often measured in terms of Sobolev, H{\"o}lder, Besov or Lipschitz spaces, etc. However, sometimes a gain of regularity can also be expressed just in terms of Lebesgue spaces, by passing from a…
We consider a quasilinear parabolic stochastic partial differential equation driven by a multiplicative noise and study regularity properties of its weak solution satisfying classical a priori estimates. In particular, we determine…
Solutions to nonlinear integro-differential systems are regular outside a negligible closed subset whose Hausdorff dimension can be explicitly bounded from above. This subset can be characterized using quantitative, universal energy…
In this article, we prove the local $C^{0,\alpha}$ regularity and provide $C^{0,\alpha}$ estimates for viscosity solutions of fully nonlinear, possibly degenerate, elliptic equations associated to linear or nonlinear Neumann type boundary…
We examine $L^p$-viscosity solutions to fully nonlinear elliptic equations with bounded-measurable ingredients. By considering $p_0<p<d$, we focus on gradient-regularity estimates stemming from nonlinear potentials. We find conditions for…
Path-dependent PDEs (PPDEs) are natural objects to study when one deals with non Markovian models. Recently, after the introduction of the so-called pathwise (or functional or Dupire) calculus (see [15]), in the case of finite-dimensional…
We introduce a class of second order backward stochastic differential equations and show relations to fully non-linear parabolic PDEs. In particular, we provide a stochastic representation result for solutions of such PDEs and discuss Monte…
We consider nonlinear integro-differential equations, like the ones that arise from stochastic control problems with purely jump L\`evy processes. We obtain a nonlocal version of the ABP estimate, Harnack inequality, and interior…
The aim of this note is to study the Cauchy problem for the 2D Euler equations under very low regularity assumptions on the initial datum. We prove propagation of regularity of logarithmic order in the class of weak solutions with $L^p$…