Second Derivatives Of Solutions Research Articles

Geometry and Conservation Laws for a Class of Second-Order Parabolic Equations II: Conservation Laws

I consider the existence and structure of conservation laws for the general class of evolutionary scalar second-order differential equations with parabolic symbol. First I calculate the linearized characteristic cohomology for such equations. This provides an auxiliary differential equation satisfied by the conservation laws of a given parabolic equation. This is used to show that conservation laws for any evolutionary parabolic equation depend on at most second derivatives of solutions. As a corollary, it is shown that the only evolutionary parabolic equations with at least one non-trivial conservation law are of Monge-Ampère type.

Geometry and conservation laws for a class of second-order parabolic equations I: Geometry

I consider the geometry of the general class of scalar 2nd-order differential equations with parabolic symbol, including non-linear and non-evolutionary parabolic equations. After defining the appropriate G-structure to model parabolic equations, I apply Cartan techniques to determine local geometric invariants (quantities invariant up to a generalized change of variables). One family of invariants gives a geometric characterization for parabolic equations of Monge–Ampère type. A second family of invariants determines when a parabolic equation has a local choice of coordinates putting it in evolutionary form.In addition to their intrinsic interest, these results are applied in a follow up paper on the conservation laws of parabolic equations. It is shown there that conservation laws for any evolutionary parabolic equation depend on at most second derivatives of solutions. As a corollary, the only evolutionary parabolic equations with at least one non-trivial conservation law are of Monge–Ampère type.

Regularity and weak comparison principles for double phase quasilinear elliptic equations

We consider the Euler equation of functionals involving a term of the form \begin{document}$ \int_{\Omega}(| \nabla u|^p+a(x)| \nabla u|^q) \,{ {\rm{d}}} x, $\end{document} with \begin{document}$ 1 and \begin{document}$ a(x)\geq 0 $\end{document} . We prove weak comparison principle and summability results for the second derivatives of solutions.

A note on the boundary regularity of solutions to quasilinear elliptic equations

We study the summability up to the boundary of the second derivatives of solutions to a class of Dirichlet boundary value problems involving thep-Laplace operator. Our results are meaningful for the cases when the Hopf’s Lemma cannot be applied to ensure that there are no critical points of the solution on the boundary of the domain.

Some remarks on the $L^p$ regularity of second derivatives of solutions to non-divergence elliptic equations and the Dini condition

In this note we prove an end-point regularity result on the L^p integrability of the second derivatives of solutions to non-divergence form uniformly elliptic equations whose second derivatives are a priori only known to be integrable. The main assumption on the elliptic operator is the Dini continuity of the coefficients. We provide a counterexample showing that the Dini condition is somehow optimal. We also give a counterexample related to the BMO regularity of second derivatives of solutions to elliptic equations. These results are analogous to corresponding results for divergence form elliptic equations in [3, 15].

Regularity and comparison principles for p-Laplace equations with vanishing source term

We prove some sharp estimates on the summability properties of the second derivatives of solutions to the equation –Δpu = f(x), under suitable assumptions on the source term. As an application, we deduce some strong comparison principles for the p-Laplacian, in the case of vanishing source terms.

A note on interior $$W^{2,1+ \varepsilon }$$ estimates for the Monge–Ampère equation

By a variant of the techniques introduced by the first two authors in De Philippis and Figalli (Invent Math 2012) to prove that second derivatives of solutions to the Monge–Ampère equation are locally in $$L\log L$$ , we obtain interior $$W^{2,1+\varepsilon }$$ estimates.

Regularity result for nondivergence equations with unbounded coefficients

The aim of this paper is to establish a higher integrability result for the second derivatives of solutions to nondivergence elliptic equations of the type $\sum_{i,j}^n a_{ij} \frac{\partial^2 u}{\partial x_i \partial x_j} =h$. The matrix coefficient $A(x)=[a_{ij}(x)]_{ij}$ is assumed to belong to the space $BMO$.

Interior C 2,α Regularity for Potential Functions in Optimal Transportation

In this paper we study the continuity of second derivatives of solutions to the Monge–Ampère equation arising in optimal transportation. We obtain Hölder and more general continuity estimates for second derivatives, when the inhomogeneous term is Hölder and Dini continuous, together with corresponding regularity results for potentials.

On the second boundary value problem for Monge-Ampère type equations and optimal transportation

This paper is concerned with the existence of globally smooth so- lutions for the second boundary value problem for certain Monge-Amp` ere type equations and the application to regularity of potentials in optimal transportation. In particular we address the fundamental issue of determining conditions on costs and domains to ensure that optimal mappings are smooth diffeomorphisms. The cost functions satisfy a weak form of the condition (A3), which was introduced in a recent paper with Xi-nan Ma, in conjunction with interior regularity. Our condition is optimal and includes the quadratic cost function case of Caffarelli and Urbas as well as the various examples in our previous work. The approach is through the derivation of global estimates for second derivatives of solutions.

Sharp regularity results on second derivatives of solutions to the Monge‐Ampère equation with VMO type data

AbstractWe establish interior estimates for Lp‐norms, Orlicz norms, and mean oscillation of second derivatives of solutions to the Monge‐Ampère equation det D2u = f(x) with zero boundary value, where f(x) is strictly positive, bounded, and satisfies a VMO‐type condition. These estimates develop the regularity theory of the Monge‐Ampère equation in VMO‐type spaces. Our Orlicz estimates also sharpen Caffarelli's celebrated W2, p‐estimates. © 2008 Wiley Periodicals, Inc.

A higher integrability result for nondivergence elliptic equations

The aim of this paper is to establish a higher integrability result of the second derivatives of solutions to nondivergence elliptic equations of the type � n i,j aij ∂ 2 u ∂xi∂xj = h. We assume that the coefficients aij are bounded and have small BMO-norm.

On the mean oscillation of the Hessian of solutions to the Monge–Ampère equation

We give interior a priori estimates for the mean oscillation of second derivatives of solutions to the Monge–Ampère equation det D 2 u = f ( x ) with zero boundary values, where f ( x ) is a non-Dini continuous function. If the modulus of continuity of f ( x ) is φ ( r ) such that lim r → 0 φ ( r ) log ( 1 / r ) = 0 , then D 2 u ∈ VMO .

Estimates in the generalized Campanato-John-Nirenberg spaces for fully nonlinear elliptic equations

We establish estimates in BMO and Campanato-John-Nirenberg spaces BMO$\sb \psi$ for the second derivatives of solutions to the fully nonlinear elliptic equation $F(D\sp 2u,x)=f(x)$.

Real analytic solutions of parabolic equations with time-measurable coefficients

We use Bernstein’s technique to show that for any fixed t t , strong solutions u ( t , x ) u(t,x) of the uniformly parabolic equation L u := a i j ( t ) u x i x j − u t = 0 Lu:=a^{ij}(t)u_{x_{i}x_{j}}-u_{t}=0 in Q Q are real analytic in Q ( t ) = { x : ( t , x ) ∈ Q } Q(t)=\{x:(t,x)\in Q\} . Here, Q ⊂ R d + 1 Q\subset \mathbb {R}^{d+1} is a bounded domain and the coefficients a i j ( t ) a^{ij}(t) are measurable. We also use Bernstein’s technique to obtain interior estimates for pure second derivatives of solutions of the fully nonlinear, uniformly parabolic, concave equation F ( D 2 u , t ) − u t = 0 F(D^{2}u,t)-u_{t}=0 in Q Q , where F F is measurable in t t .

A superconvergent method for approximating the bending moment of elastic beams with hysteresis damping

In this paper we introduce a method for finding improved approximations to second derivatives of solutions to certain initial-boundary value problems related to elastic beam models. These approximations are based on post-processing finite element solutions of the integro–partial-differential equation involved. Illustrative numerical examples for this method and its applications to an inverse problem of such models are provided.

Unimprovable Integral Estimate for Solutions of the Dirichlet Problem for Quasilinear Elliptic Equations of the Second Order in a Neighborhood of an Edge

We obtain an exact estimate for the second derivatives of solutions (in the weight Sobolev norm) of the Dirichlet problem for quasilinear second-order elliptic equations of nondivergence type in a neighborhood of an edge of a domain.

Prior bounds on generalized derivatives of solutions of linear elliptical equations and systems of second order

We derive new prior bounds on generalized first and second derivatives of solutions for the equations $$(\lambda - d \circ \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a} \circ d)u = f, (\lambda - d \circ a \circ u)\bar u = \mathop f\limits^--- ,\forall f,\mathop f\limits^--- \in L^1 \cup L^\infty \cup C^\infty , \lambda > 0$$ with smooth coefficients in Rl. The bounds are established in the scale of the spaces L∞ ∩ W1 2p ∩ W 2 p and depend on the structure of the equations.

Maximum principles related to the second derivatives of solutions of elliptic equations in the plane

On considere des equations elliptiques non lineaires a deux variables F(D 2 u)=0. On demontre que D 2 u applique des ensembles ouverts dans des ensembles ouverts, s'il n'est pas constant. On donne des principes de maximum pour des fonctions de D 2 u

A priori estimates of the second derivatives of solutions to nonlinear second-order equations on the boundary of the domain

Local estimates on the boundary of a domain for the gradients of solutions of second-order quasilinear elliptic equations are constructed. These estimates are applied to establish local estimates on the boundary for the second derivatives of solutions of a certain class of second-order nonlinear elliptic equations.