We report on new techniques and results in the regularity theory of general non-uniformly elliptic variational integrals. By means of a new potential theoretic approach we reproduce, in the non-uniformly elliptic setting, the optimal criteria for Lipschitz continuity known in the uniformly elliptic one and provide a unified approach between non-uniformly and uniformly elliptic problems.
For the singularly perturbed system \[Δu_{i,β}=βu_{i,β}\sum_{j\neq i}u_{j,β}^2, \quad 1\leq i\leq N,\] we prove that flat interfaces are uniformly Lipschitz. As a byproduct of the proof we also obtain the optimal lower bound near the flat interfaces, \[\sum_iu_{i,β}\geq cβ^{-1/4}.\]
We discuss conditions for Lipschitz and C^1 regularity for a uniformly elliptic equation in divergence form with coefficients that were introduced by Gilbarg & Serrin. In particular, we find cases where Lipschitz or C^1 regularity holds but the coefficients are not Dini continuous, or do not even have Dini mean oscillation. The form of the coefficients also enables us to obtain specific conditions and examples for which there exists a weak solution that is not Lipschitz continuous.
We consider second order uniformly elliptic operators of divergence form in $\R^{d+1}$ whose coefficients are independent of one variable. Under the Lipschitz condition on the coefficients we characterize the domain of the Poisson operators and the Dirichlet-Neumann maps in the Sobolev space $H^s(\R^d)$ for each $s\in [0,1]$. Moreover, we also show a factorization formula for the elliptic operator in terms of the Poisson operator.
We provide a general approach to Lipschitz regularity of solutions for a large class of vector-valued, nonautonomous variational problems exhibiting nonuniform ellipticity. The functionals considered here range amongst those with unbalanced polynomial growth conditions to those with fast, exponential type growth. The results obtained are sharp with respect to all the data considered and yield new, optimal regularity criteria even in the classical uniformly elliptic case. We give a classification of different types of nonuniform ellipticity, accordingly identifying suitable conditions to get re…
We obtain an error estimate between viscosity solutions and δ-viscosity solutions of nonhomogeneous fully nonlinear uniformly elliptic equations. The main assumption, besides uniform ellipticity, is that the nonlinearity is Lipschitz-continuous in space with linear growth in the Hessian. We also establish a rate of convergence for monotone and consistent finite difference approximation schemes for such equations.
We consider vector-valued solutions to a linear transmission problem, and we prove that Lipschitz-regularity on one phase is transmitted to the next phase. More exactly, given a solution $u:B_1\subset \mathbb{R}^n \to \mathbb{R}^m$ to the elliptic system \begin{equation*} \mbox{div} ((A + (B-A)χ_D )\nabla u) = 0 \quad \text{in }B_1, \end{equation*} where $A$ and $B$ are Dini continuous, uniformly elliptic matrices, we prove that if $\nabla u \in L^{\infty} (D)$ then $u$ is Lipschitz in $B_{1/2}$. A similar result is also derived for the parabolic counterpart of this problem.
We study local regularity properties of local minimizer of scalar integral functionals of the form $$\mathcal F[u]:=\int_ΩF(\nabla u)-f u\,dx$$ where the convex integrand $F$ satisfies controlled $(p,q)$-growth conditions. We establish Lipschitz continuity under sharp assumptions on the forcing term $f$ and improved assumptions on the growth conditions on $F$ with respect to the existing literature. Along the way, we establish an $L^\infty$-$L^2$-estimate for solutions of linear uniformly elliptic equations in divergence form which is optimal with respect to the ellipticity contrast of the coe…
We consider uniformly elliptic operators with Dirichlet or Neumann homogeneous boundary conditions on a domain $Ω$ in ${\mathbb{R}}^N$. We consider deformations $ϕ(Ω)$ of $Ω$ obtained by means of a locally Lipschitz homeomorphism $ϕ$ and we estimate the variation of the eigenfunctions and eigenvalues upon variation of $ϕ$. We prove general stability estimates without using uniform upper bounds for the gradients of the maps $ϕ$. As an application, we obtain estimates on the rate of convergence for eigenvalues and eigenfunctions when a domain with an outward cusp is approximated by a sequence of…
We define a random step size tug-of-war game, and show that the gradient of a value function exists almost everywhere. We also prove that the gradients of value functions are uniformly bounded and converge weakly to the gradient of the corresponding $p$-harmonic function. Moreover, we establish an improved Lipschitz estimate when boundary values are close to a plane. Such estimates are known to play a key role in higher regularity theory of partial differential equations. The proofs are based on cancellation and coupling methods as well as improved version of the cylinder walk argument.
In this paper, we analyze the behavior of a family of solutions of a nonlinear elliptic equation with nonlinear boundary conditions, when the boundary of the domain presents a highly oscillatory behavior which is uniformly Lipschitz and nonlinear terms are concentrated in a region which neighbors the boundary domain. We prove that this family of solutions converges to the solutions of a limit problem in H^1 , an elliptic equation with nonlinear boundary conditions which captures the oscillatory behavior of the boundary and whose nonlinear terms are transformed into a flux condition on the boun…
In this paper, we provide suitable adaptations of the "weak version of Bernstein method" introduced by the first author in 1991, in order to obtain Lipschitz regularity results and Lipschitz estimates for nonlinear integro-differential elliptic and parabolic equations set in the whole space. Our interest is to obtain such Lipschitz results to possibly degenerate equations, or to equations which are indeed "uniformly el-liptic" (maybe in the nonlocal sense) but which do not satisfy the usual "growth condition" on the gradient term allowing to use (for example) the Ishii-Lions' method. We treat …
The Cauchy problem for a multidimensional linear transport equation with discontinuous coefficient is investigated. Provided the coefficient satisfies a one-sided Lipschitz condition, existence, uniqueness and weak stability of solutions are obtained for either the conservative backward problem or the advective forward problem by duality. Specific uniqueness criteria are introduced for the backward conservation equation since weak solutions are not unique. A main point is the introduction of a generalized flow in the sense of partial differential equations, which is proved to have unique jacob…
The present paper concerns the well-posedness of the Cauchy problem for microlocally symmetrizable hyperbolic systems whose coefficients and symmetrizer are log-Lipschitz continuous, uniformly in time and space variables. For the global in space problem we establish energy estimates with finite loss of derivatives, which is linearly increasing in time. This implies well-posedness in $H^\infty$, if the coefficients enjoy enough smoothness in $x$. From this result, by standard arguments (i.e. extension and convexification) we deduce also local existence and uniqueness. A huge part of the analysi…
In this paper we obtain an energy estimate for a complete strictly hyperbolic operator with second order coefficients satisfying a log-Zygmund-continuity condition with respect to $t$, uniformly with respect to $x$, and a log-Lipschitz-continuity condition with respect to $x$, uniformly with respect to $t$.
We use the method of layer potentials to study the $R_2$ Regularity problem and the $D_2$ Dirichlet problem for second order elliptic equations of the form $\mathcal{L}u=0$, with lower order coefficients, in bounded Lipschitz domains. For $R_2$ we establish existence and uniqueness assuming that $\mathcal{L}$ is of the form $\mathcal{L}u=-\text{div}(A\nabla u+bu)+c\nabla u+du$, where the matrix $A$ is uniformly elliptic and Hölder continuous, $b$ is Hölder continuous, and $c,d$ belong to Lebesgue classes and they satisfy either the condition $d\geq\text{div}b$, or $d\geq\text{div}c$ in the sen…
We consider the minimization problem of an anisotropic energy in classes of $d$-rectifiable varifolds in $\mathbb R^n$, closed under Lipschitz deformations and encoding a suitable notion of boundary. We prove that any minimizing sequence with density uniformly bounded from below converges (up to subsequences) to a $d$-rectifiable varifold. Moreover, the limiting varifold is integral, provided the minimizing sequence is made of integral varifolds with uniformly locally bounded anisotropic first variation.
This paper addresses the problem of wellposedness of non-autonomous linear evolution equations $\dot x = A(t)x$ in uniformly convex Banach spaces. We assume that $A(t):D \subset X\to X$, for each $t$ is the generator of a quasi-contractive $C_0$-group where the domain $D$ and the growth exponent are independent of $t$. Well-posedness holds provided that $t\mapsto A(t)y$ is Lipschitz for all $y\in D$. Hölder continuity of degree $α<1$ is not sufficient and the assumption of uniform convexity cannot be dropped.
A subset $A$ of the state space is called uniformly globally weakly attractive if for any neighborhood $S$ of $A$ and any bounded subset $B$ there is a uniform finite time $τ$ so that any trajectory starting in $B$ intersects $S$ within the time not larger than $τ$. We show that practical uniform global asymptotic stability (pUGAS) is equivalent to the existence of a bounded uniformly globally weakly attractive set. This result is valid for a wide class of distributed parameter systems, including time-delay systems, switched systems, many classes of PDEs and evolution differential equations in…
