Related papers: Fully nonlinear parabolic fixed transmission probl…
We develop the regularity theory of viscosity solutions to transmission problems for fully nonlinear second order uniformly elliptic equations. Our results give a complete theory of existence, uniqueness, comparison principle, and…
We examine a free transmission problem driven by fully nonlinear elliptic operators. Since the transmission interface is determined endogeneously, our analysis is two-fold: we study the regularity of the solutions and some geometric…
We prove $C^{1, \alpha}$ regularity (in the parabolic sense) for the viscosity solution of a boundary obstacle problem with a fully nonlinear parabolic equation in the interior. Following the method which was first introduced for the…
We prove that boundary value problems for fully nonlinear second-order parabolic equations admit $L_{p}$-viscosity solutions, which are in $C^{1+\alpha}$ for an $\alpha\in(0,1)$. The equations have a special structure that the "main" part…
In this paper we prove that solutions to a transmission problem degenerating on the interface are H\"older differentiable up to the interface with universal estimates. Furthermore, we obtain a sharper pointwise $C^{1,\alpha(\cdot)}$ with…
We consider a transmission problem consisting of a semilinear parabolic equation in a general non-smooth setting with emphasis on rough interfaces which bear a fractal-like geometry and nonlinear dynamic (possibly, nonlocal)\ boundary…
We consider a class of elliptic and parabolic problems, featuring a specific nonlocal operator of fractional-laplacian type, where integration is taken on variable domains. Both elliptic and parabolic problems are proved to be uniquely…
We study a free transmission problem driven by degenerate fully nonlinear operators. Our first result concerns the existence of solutions to the associated Dirichlet problem. By framing the equation in the context of viscosity inequalities,…
We study an equation governed by a discontinuous fully nonlinear operator. Such discontinuities are solution-dependent, which introduces a free boundary. Working under natural assumptions, we prove the existence of $L^p$-viscosity and…
We study the regularity of the viscosity solution to the fully nonlinear parabolic thin obstacle problem. In particular, we prove that the solution is local $H^{1+\alpha}$ on each side of the smooth obstacle, for some small $\alpha>0.$…
In this paper, we obtain the boundary pointwise $C^{1,\alpha}$ and $C^{2,\alpha}$ regularity for viscosity solutions of fully nonlinear elliptic equations. I.e., If $\partial \Omega$ is $C^{1,\alpha}$ (or $C^{2,\alpha}$) at $x_0\in \partial…
This paper is concerned with the Minkowski convolution of viscosity solutions of fully nonlinear parabolic equations. We adopt this convolution to compare viscosity solutions of initial-boundary value problems in different domains. As a…
We present a new stability result for viscosity solutions of fully nonlinear parabolic equations which allows to pass to the limit when one has only weak convergence in time of the nonlinearities.
We investigate the $C^{1+\alpha}$-regularity of solutions of parabolic equations $\partial_{t}v+H(v,Dv,D^{2}v,t,x)=0$. Our main result says that under rather general assumptions there exist $C$-viscosity and $L_{p}$-viscosity solutions…
We study a one-dimensional nonlinear hyperbolic-parabolic initial boundary value problem occurring in the theory of thermoelasticity. We prove existence and uniqueness of the local-in-time strong solution. Also, some global-in-time weak…
In this paper, we prove boundary pointwise $C^{k,\alpha}$ regularity for any $k\geq 1$ for fully nonlinear parabolic equations. As an application, we give a direct and short proof of the higher regularity of the free boundaries in…
The interaction between a viscous fluid and an elastic solid is modeled by a system of parabolic and hyperbolic equations, coupled to one another along the moving material interface through the continuity of the velocity and traction…
We establish the interior $C^{1,\alpha}$-estimate for viscosity solutions of degenerate/singular fully nonlinear parabolic equations $$u_t = |Du|^{\gamma}F(D^2u) + f.$$ For this purpose, we prove the well-posedness of the regularized…
We prove the $W^{1,2}_{p}$-solvability of second order parabolic equations in nondivergence form in the whole space for $p\in (1,\infty)$. The leading coefficients are assumed to be measurable in one spatial direction and have vanishing…
In this paper we study linear parabolic equations on a finite oriented starshaped network; the equations are coupled by transmission conditions set at the inner node, which do not impose continuity on the unknown. We consider this problem…