In this paper, we develop systematically the pointwise regularity for viscosity solutions of fully nonlinear elliptic equations in general forms. In particular, the equations with quadratic growth (called natural growth) in the gradient are covered. We obtain a series of interior and boundary pointwise $C^{k,α}$ regularity ($k\geq 1$ and $0<α<1$). In addition, we also derive the pointwise $C^k$ regularity ($k\geq 1$) and $C^{k,\mathrm{lnL}}$ regularity ($k\geq 0$), which correspond to the end points $α=0$ and $α=1$ respectively. Some regularity results are new even for the linear equations. Moreover, the minimum requirements are imposed to obtain above regularity and our proofs are simple.
In this paper, we develop a series of boundary pointwise regularity for Dirichlet problems and oblique derivative problems. As applications, we give direct and simple proofs of the higher regularity of the free boundaries in obstacle-type problems and one phase problems.
In this paper, we prove boundary pointwise $C^{k,α}$ 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 obstacle-type problems.
In this paper, we prove a boundary pointwise regularity for fully nonlinear elliptic equations on cones. In addition, based on this regularity, we give simple proofs of the Liouville theorems on cones.
In this paper, we obtain the boundary pointwise $C^{1,α}$ and $C^{2,α}$ regularity for viscosity solutions of fully nonlinear elliptic equations. I.e., If $\partial Ω$ is $C^{1,α}$ (or $C^{2,α}$) at $x_0\in \partial Ω$, the solution is $C^{1,α}$ (or $C^{2,α}$) at $x_0$. Our results are new even for the Laplace equation. Moreover, our proofs are simple.
We consider the pointwise in space Lp-type regularity for elliptic and parabolic equations of order m in Rn. We provide pointwise Schauder estimates for the general range of Lp exponents, extending previous results from p > n/m to 1 < p < n/m.
In this paper, we prove the pointwise boundary differentiability for viscosity solutions of fully nonlinear elliptic equations. This generalizes the previous related results for linear equations. The geometrical conditions in this paper are pointwise and more general than before. Moreover, our proofs are relatively simple.
A sharp pointwise differential inequality for vectorial second-order partial differential operators, with Uhlenbeck structure, is offered. As a consequence, optimal second-order regularity properties of solutions to nonlinear elliptic systems in domains in $\mathbb{R}^n$ are derived. Both local and global estimates are established. Minimal assumptions on the boundary of the domain are required for the latter. In the special case of the $p$-Laplace system, our conclusions broaden the range of the admissible values of the exponent $p$ previously known.
We introduce a new approach to obtaining pointwise estimates for solutions of elliptic boundary value problems when the operator being considered satisfies a certain type of weighted integral inequalities. The method is illustrated on several examples, including a scalar second-order elliptic equation, the 3D Lamé system, and a scalar higher-order elliptic equation. The techniques can be extended to other elliptic boundary value problems provided that the corresponding weighted integral inequalities are satisfied.
We investigate the long time behavior of a system of viscoelastic particles modeled with the homogeneous Boltzmann equation. We prove the existence of a universal Maxwellian intermediate asymptotic state and explicit the rate of convergence towards it. Exponential lower pointwise bounds and propagation of regularity are also studied. These results can be seen as the generalization of several classical facts holding for the pseudo-Maxwellian and constant normal restitution models.
We study the parabolic obstacle problem $$\lap u-u_t=fχ_{\{u>0\}}, \quad u\geq 0,\quad f\in L^p \quad \mbox{with}\quad f(0)=1$$ and obtain two monotonicity formulae, one that applies for general free boundary points and one for singular free boundary points. These are used to prove a second order Taylor expansion at singular points (under a pointwise Dini condition), with an estimate of the error (under a pointwise double Dini condition). Moreover, under the assumption that $f$ is Dini continuous, we prove that the set of regular points is locally a (parabolic) $C^1$-surface and that the set o…
We obtain sharp pointwise estimates for positive solutions to the equation $-Lu+Vu^q=f$, where $L$ is an elliptic operator in divergence form, $q\in\mathbb{R}\setminus \{0\}$, $f\geq 0$ and $V$ is a function that may change sign, in a domain $Ω$ in $\mathbb{R}^{n}$, or in a weighted Riemannian manifold.
The study of reflector surfaces in geometric optics necessitates the analysis of certain nonlinear equations of Monge-Ampère type known as generated Jacobian equations. These equations, whose general existence theory has been recently developed by Trudinger go beyond the framework of optimal transport. We obtain pointwise estimates for weak solutions of such equations under a condition analogous to the A3w condition of Ma, Trudinger and Wang. Estimates of this type have played an important role in the regularity theory for optimal transport maps and were previously unknown in this context, inc…
This paper establishes pointwise estimates up to boundary for the gradient of weak solutions to a class of very singular quasilinear elliptic equations with mixed data: \begin{cases} -\operatorname{div}(A(x,D u))=g-\operatorname{div} f \quad & \mathrm{in} \quad Ω\\ u= 0 \quad & \text{on} \ \partial Ω, \end{cases} where $Ω\subset \mathbb{R}^n$ is sufficiently flat in the sense of Reifenberg.
We give pointwise gradient bounds for solutions of (possibly non-uniformly) elliptic partial differential equations in the entire Euclidean space. The operator taken into account is very general and comprises also the singular and degenerate nonlinear case with non-standard growth conditions. The sourcing term is also allowed to have a very general form, depending on the space variables, on the solution itself, on its gradient, and possibly on higher order derivatives if additional structural conditions are satisfied.
We demonstrate a measure theoretical approach to the local regularity of weak supersolutions to elliptic and parabolic equations in divergence form. In the first part, we show that weak supersolutions become lower semicontinuous after redefinition on a set of measure zero. The proof relies on a general principle, i.e. the De Giorgi type lemma, which offers a unified approach for a wide class of elliptic and parabolic equations, including an anisotropic elliptic equation, the parabolic $p$-Laplace equation, and the porous medium equation. In the second part, we shall show that for parabolic pro…
In this paper, we give pointwise geometric conditions on the boundary which guarantee the differentiability of the solution at the boundary. Precisely, the geometric conditions are two parts: the proper blow up condition (see Definition 1) and the exterior Dini hypersurface condition (see Definition 2). If $Ω$ satisfies this two conditions at $x_0\in\partial Ω$, the solution is differentiable at $x_0$. Furthermore, counterexamples show that the conditions are optimal (see Remark 3 and the counterexample in Section 2).
This article is about the $\mathbb{Z}^d$-periodic Green function $G_n(x,y)$ of the multiscale elliptic operator $Lu=-{\rm div}\left( A(n\cdot) \cdot \nabla u \right)$, where $A(x)$ is a $\mathbb{Z}^d$-periodic, coercive, and Hölder continuous matrix, and $n$ is a large integer. We prove here pointwise estimates on $G_n(x,y)$, $\nabla_x G_n(x,y)$, $\nabla_y G_n(x,y)$ and $\nabla_x \nabla_y G_n(x,y)$ in dimensions $d \geq 2$. Moreover, we derive an explicit decomposition of this Green function, which is of independent interest. These results also apply for systems.
We establish global pointwise bounds for the Green's matrix for divergence form, second order elliptic systems in a domain under the assumption that weak solutions of the system vanishing on a portion of the boundary satisfy a certain local boundedness estimate. Moreover, we prove that such a local boundedness estimate for weak solutions of the system is equivalent to the usual global pointwise bound for the Green's matrix. In the scalar case, such an estimate is a consequence of De Giorgi-Moser-Nash theory and holds for equations with bounded measurable coefficients in arbitrary domains. In t…
We study quasilinear elliptic equations of the type $$-Δ_pu=σ\, u^q \quad \text{in} \, \, \, \mathbb{R}^n,$$ where $Δ_p u=\nabla \cdot(\nabla u |\nabla u|^{p-2})$ is the $p$-Laplacian (or a more general $\mathcal{A}$-Laplace operator $\text{div} \, \mathcal{A}(x, \nabla u)$), $0<q < p-1$, and $σ\ge 0$ is an arbitrary locally integrable function or measure on $\mathbb{R}^n$. We obtain necessary and sufficient conditions for the existence of positive solutions (not necessarily bounded) which satisfy global pointwise estimates of Brezis-Kamin type given in terms of Wolff potentials. Similar probl…
We study the behavior near the origin of $C^2$ positive solutions $u(x)$ and $v(x)$ of the system $0\leq -Δu\leq f(v)$ $0\leq -Δv\leq g(u)$ in $B_1(0)\backslash\{0\}$ where $f,g:(0,\infty)\to (0,\infty)$ are continuous functions. We provide optimal conditions on $f$ and $g$ at $\infty$ such that solutions of this system satisfy pointwise bounds near the origin. In dimension $n=2$ we show that this property holds if $\log^+ f$ or $\log^+g$ grow at most linearly at infinity. In dimension $n\geq 3$ and under the assumption $f(t)=O(t^λ)$, $g(t)=O(t^σ)$ as $t\to \infty$, ($λ, σ\geq 0$), we obtain a…
We obtain pointwise lower bounds for heat kernels of higher order differential operators with Dirichlet boundary conditions on bounded domains in $\R^N$. The bounds exhibit explicitly the nature of the spatial decay of the heat kernel close to the boundary. We make no smoothness assumptions on our operator coefficients which we assume only to be bounded and measurable.
In the present paper, we investigate the regularity and symmetry properties of weak solutions to semilinear elliptic equations which are locally stable.
Pointwise arbitrarily high-order interior estimates for mixed local and nonlocal elliptic equations
School of Mathematics and Statistics, Xinyang Normal University（信阳师范大学数学与统计学院）；School of Mathematics, South China University of Technology（华南理工大学数学学院）；School of Mathematical Sciences, Fudan University（复旦大学数学科学学院）
We construct the Green function for second-order elliptic equations in non-divergence form when the mean oscillations of the coefficients satisfy the Dini condition. We show that the Green's function is BMO in the domain and establish logarithmic pointwise bounds. We also obtain pointwise bounds for first and second derivatives of the Green's function.
We establish an optimal C^{1,α}-regularity for viscosity solutions of degenerate/singular fully nonlinear elliptic equations by finding minimal regularity requirements on the associated operator.
We derive a priori bounds for positive supersolutions of $ - Δ_{p} u = ρ(x) f(u) $, where $p>1$ and $Δ_{p}$ is the $p$-Laplace operator, in a smooth bounded domain of $R^{N}$ with zero Dirichlet boundary conditions. We apply the results to nonlinear elliptic eigenvalue problem $ - Δ_{p} u = λf(u) $, with Dirichlet boundary condition, where $ f $ is a nondecreasing continuous differentiable function on $[0,\infty]$ such that $ f(0) > 0 $, $ f(t)^{\frac{1}{p-1}} $ is superlinear at infinity, and give sharp upper and lower bounds for the extremal parameter $ λ_{p}^{*} $. In particular, we conside…
In this note we study the global regularity in the Morrey spaces for the second derivatives for the strong solutions of non variational elliptic equations.
We study pointwise convergence of the solution to the elastic wave equation to the initial data which lies in the Sobolev spaces. We prove that the solution converges along every lines to the initial data almost everywhere whenever the initial regularity is greater than one half. We show this is almost optimal.
We obtain an explicit Hölder regularity result for viscosity solutions of a class of second order fully nonlinear equations leaded by operator that are neither convex/concave nor uniformly elliptic.
In this work we study regularity properties of solutions to fractional elliptic problems with mixed Dirichlet-Neumann boundary data when dealing with the Spectral Fractional Laplacian.
We prove a $C^{1,α}$ 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 of a special class of nonlinear operators.
We prove a Meyers type regularity estimate for approximate solutions of second order elliptic equations obtained by Galerkin methods. The proofs rely on interpolation results for Sobolev spaces on graphs. Estimates for second order elliptic operators on rather general graphs are also obtained.
In this paper, we study the regularity of weak solutions to a class of nonlocal elliptic equations in Bessel potential spaces $H^{s,p}$. Our main results can be seen as an extension of the well-known $W^{1,p}$ regularity theory for local second-order elliptic equations in divergence form to the nonlocal setting.
We establish sharp global regularity results for solutions to nonhomogeneous, nonunifomrly elliptic systems with zero boundary conditions. In particular, we obtain everywhere Lipschitz continuity under borderline Lorentz assumptions on the forcing term, thus positively settling the optimality issue raised in \cite{bdms}.
In this paper,we consider the solutions of the non-homogeneous elliptic obstacle problems with Orlicz growth involving measure data. We first establish the pointwise estimates of the approximable solutions to these problems via fractional maximal operators. As a result, we obtain pointwise and oscillation estimates for the gradients of solutions by the non-linear Wolff potentials, and these yield results on $C^{1,α}$-regularity of solutions.
In this paper we study the existence and partial regularity of weak solutions to an elliptic-parabolic system that models the single-phase miscible displacement of one incompressible fluid by another in a porous media. The system is singular and involves quadratic nonlinearity. A partial regularity result is obtained via the Fefferman-Stein inequality.
We establish partial regularity result for vector-valued solutions to second order elliptic system in divergence form. The coefficients safisfies Dini condition respect to $(x,u)$ with growth order lager than 2. We prove $C^1$-regularity of the solutions outside of singular set.
