Multiple integrals in the calculus of variations and nonlinear elliptic systems

Differential equations with non-standard growth have been a very active field of investigation in recent years. In this survey we present an overview of the field, as well as several of the most important results. We consider both existence and regularity questions. Finally, we provide a comprehensive list of papers published to date.

Overview of differential equations with non-standard growth

Multiple Integrals In The Calculus Of Variations And Nonlinear Elliptic Systems

Weakly Differentiable Functions

We present pointwise gradient bounds for solutions to $p$-Laplacean type non-homogeneous equations employing non-linear Wolff type potentials, and then prove similar bounds, via suitable caloric potentials, for solutions to parabolic equations

Gradient estimates via non-linear potentials

Abstract We establish the natural Calderón and Zygmund theory for solutions of elliptic and parabolic obstacle problems involving possibly degenerate operators in divergence form of p-Laplacian type, and proving that the (spatial) gradient of solutions is as integrable as that of the assigned obstacles. We also include an existence and regularity theorem where obstacles are not necessarily considered to be non-increasing in time.

Degenerate problems with irregular obstacles

We consider the evolution problem associated with a convex integrand ${f : \mathbb{R}^{Nn}\to [0,\infty)}$ satisfying a non-standard p, q-growth assumption. To establish the existence of solutions we introduce the concept of variational solutions. In contrast to weak solutions, that is, mappings ${u\colon \Omega_T \to \mathbb{R}^n}$ which solve 

$$ \partial_tu-{\rm div} Df(Du)=0 $$

weakly in ${\Omega_T}$, variational solutions exist under a much weaker assumption on the gap q − p. Here, we prove the existence of variational solutions provided the integrand f is strictly convex and 

$$\frac{2n}{n+2} < p \le q < p+1.$$

These variational solutions turn out to be unique under certain mild additional assumptions on the data. Moreover, if the gap satisfies the natural stronger assumption 

$$ 2\le p \le q < p+ {\rm min}\big \{1,\frac{4}{n} \big \},$$

we show that variational solutions are actually weak solutions. This means that solutions u admit the necessary higher integrability of the spatial derivative Du to satisfy the parabolic system in the weak sense, that is, we prove that 

$$u\in L^q_{\rm loc}\big(0,T; W^{1,q}_{\rm loc}(\Omega,\mathbb{R}^N)\big).$$

Parabolic Systems with p, q-Growth: A Variational Approach

The aim of the paper is twofold. On one hand the authors want to present a new technique called $p$-caloric approximation, which is a proper generalization of the classical compactness methods first developed by DeGiorgi with his Harmonic Approximation Lemma. This last result, initially introduced in the setting of Geometric Measure Theory to prove the regularity of minimal surfaces, is nowadays a classical tool to prove linearization and regularity results for vectorial problems. Here the authors develop a very far reaching version of this general principle devised to linearize general degenerate parabolic systems. The use of this result in turn allows the authors to achieve the subsequent and main aim of the paper, that is, the implementation of a partial regularity theory for parabolic systems with degenerate diffusion of the type $\partial_t u - \mathrm{div} a(Du)=0$, without necessarily assuming a quasi-diagonal structure, i.e. a structure prescribing that the gradient non-linearities depend only on the the explicit scalar quantity.

/pdf/the-regularity-of-general-parabolic-systems-with-degenerate-hvgj1hvzto.pdf

The Regularity of General Parabolic Systems With Degenerate Diffusion

http://web.math.unifi.it/users/marcellini/lavori/reprints/2014_Bogelein_Duzaar_Marcellini_J_Differential_Equations_Existence_of_evolutionary_variational_solutions_via_the_calculus_of_variations.pdf

Existence of evolutionary variational solutions via the calculus of variations

http://web.math.unifi.it/users/marcell/lavori/reprints/2013_Bogelein_Duzaar_Marcellini_J_Math_Pures_Appl_Parabolic_equations_with_p,q-growth.pdf

Verena Bögelein

Papers

Degenerate problems with irregular obstacles

Parabolic Systems with p, q-Growth: A Variational Approach

The Regularity of General Parabolic Systems With Degenerate Diffusion

Existence of evolutionary variational solutions via the calculus of variations

Parabolic equations with p, q-growth