Multiple integrals in the calculus of variations and nonlinear elliptic systems

The method of viscosity solutions for nonlinear partial differential equations (PDEs) justifies passages to limits by in effect using the maximum principle to convert to the corresponding limit problem for smooth test functions. We describe in this paper a “perturbed test function” device, which entails various modifications of the test functions by lower order correctors. Applications include homogenisation for quasilinear elliptic PDEs and approximation of quasilinear parabolic PDEs by systems of Hamilton-Jacobi equations.

The perturbed test function method for viscosity solutions of nonlinear PDE

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Analysis of the heterogeneous multiscale method for elliptic homogenization problems

An extension theorem from connected sets, and homogenization in general periodic domains

Weakly Differentiable Functions

We study the homogenization of second order quasilinear operators of the form

$$A_\varepsilon u = - div a\left( {\frac{x}{\varepsilon },u,Du} \right)$$

in Sobolev spaces H1,p (p>1). An explicit formula of the homogenized operator is given.

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On the homogenization of quasilinear divergence structure operators

We identify the exact degree of the improvement of integrability of nonnegative volume forms and, in particular, the Jacobian of orientation preserving mappings from various Orlicz-Sobolev classes. The improvement takes place when the Jacobian belongs to the Orlicz space L(Ω), where Ψ grows almost linearly, that is, t1−e ≺ Ψ(t) ≺ t for e > 0. Our results amount to the principle: further from Lloc(Ω) lower the improvement of integrability. In fact, as shown in [LZ], [Wu], [GIM], the largest improvement happens when the Jacobian is precisely in the space Lloc(Ω).

/pdf/limits-of-the-improved-integrability-of-the-volume-forms-3rpkrycv8a.pdf

Limits of the improved integrability of the volume forms

Geometric Function Theory and Nonlinear Analysis

Bi-Sobolev mappings and elliptic equations in the plane

It is proved that if the function f verifies a reverse integral inequality, then the nonincreasing rearrangement of f verifies the same inequality. As an application, a short proof of higher integrability results from reverse integral inequalities is given, by reduction to the one dimensional case.

Gioconda Moscariello

Papers

On the homogenization of quasilinear divergence structure operators

Limits of the improved integrability of the volume forms

Geometric Function Theory and Nonlinear Analysis

Bi-Sobolev mappings and elliptic equations in the plane

Higher Integrability Results.