Related papers: Global optimal regularity for variational problems…
We establish the existence and uniqueness of solutions of fully nonlinear elliptic second-order equations like $H(v,Dv,D^{2}v,x)=0$ in smooth domains without requiring $H$ to be convex or concave with respect to the second-order…
We consider the Dirichlet boundary value problem for nonlinear systems of partial differential equations with p-structure. We choose two representative cases: the "full gradient case", corresponding to a p-Laplacian, and the "symmetric…
The global existence of weak solutions of the incompressible viscoelastic flows in two spatial dimensions has been a long standing open problem, and it is studied in this paper. We show the global existence if the initial deformation…
This article has two purposes. The first is to prove solutions of the second boundary value problem for generated Jacobian equations are strictly $g$-convex. The second is to prove the global $C^3$ regularity of Aleksandrov solutions to the…
In this paper we prove global bounds on the spatial gradient of viscosity solutions to second order linear and nonlinear parabolic equations in $(0,T) \times \R^N$. Our assumptions include the case that the coefficients be both unbounded…
We establish the existence of solutions of fully nonlinear elliptic second-order equations like $H(v,Dv,D^{2}v,x)=0$ in smooth domains without requiring $H$ to be convex or concave with respect to the second-order derivatives. Apart from…
We establish the optimal regularity of viscosity solutions to \begin{equation*} u_t - x_n^\gamma \Delta u = f, \end{equation*} which arises in the regularity theory for the porous medium equation. Specifically, we prove that under the zero…
In this paper we establish optimal $C^{1,\alpha}$ regularity up to the boundary for viscosity solutions of fully nonlinear elliptic equations with double phase degeneracy law and oblique boundary conditions. The approach developed here…
This work is devoted to the studies of a Hamilton--Jacobi equation with a quadratic and degenerate Hamiltonian, which comes from the dynamics of a multipeakon in the Camassa--Holm equation. It is given by a quadratic form with a singular…
In this manuscript, we aim to establish global existence of weak solutions with higher regularity to the compressible Navier-Stokes equations under no-slip boundary conditions. Though Lions\cite{L1} and Feireisl\cite{F1} have established…
We consider Dirichlet problems for fully nonlinear mixed local-nonlocal non-translation invariant operators. For a bounded $C^2$ domain $\Omega \subset \mathbb{R}^d,$ let $u\in C(\mathbb{R}^d)$ be a viscosity solution of such Dirichlet…
This paper establishes the global in time existence of classical solutions to the 2D anisotropic Boussinesq equations with vertical dissipation. When only the vertical dissipation is present, there is no direct control on the horizontal…
We prove the optimal global regularity of nonnegative solutions to the porous medium equation in smooth bounded domains with the zero Dirichlet boundary condition after certain waiting time $T^*$. More precisely, we show that solutions are…
We prove optimal regularity for the double obstacle problem when obstacles are given by solutions to Hamilton-Jacobi equations that are not $C^2$. When the Hamilton-Jacobi equation is not $C^2$ then the standard Bernstein technique fails…
In this paper, we study the global regularity of large solutions with vacuum to the two-dimensional compressible Navier-Stokes equations on $\mathbb{T}^{2}=\mathbb{R}^{2}/\mathbb{Z}^{2}$, when the volume (bulk) viscosity coefficient $\nu$…
We study Hessian fully nonlinear uniformly elliptic equations and show that the second derivatives of viscosity solutions of those equations (in 12 or more dimensions) can blow up in an interior point of the domain. We prove that the…
In this paper, we first investigate necessary optimality conditions for problems governed by systems describing the flow of an incompressible second grade fluid. Next, we study the asymptotic behavior of the optimal solution when the…
We show that for any uniformly parabolic fully nonlinear second-order equation with bounded measurable "coefficients" and bounded "free" term in any cylindrical smooth domain with smooth boundary data one can find an approximating equation…
This paper is devoted to the study of uniform $W^{1,\frac{np}{n-p}}$- and $W^{2,p}$-estimates for viscosity solutions to fully nonlinear, uniformly elliptic, periodic homogenization problems, up to boundaries, subject to Dirichlet boundary…
Second-order optimality conditions are essential for nonsmooth optimization, where both the objective and constraint functions are Lipschitz continuous and second-order directionally differentiable. This paper provides no-gap second-order…