Isernia, Teresa (2016) Regularity results for asymptotic problems. [Tesi di dottorato]

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Item Type: Tesi di dottorato
Lingua: English
Title: Regularity results for asymptotic problems
Date: 31 March 2016
Number of Pages: 94
Institution: Università degli Studi di Napoli Federico II
Department: Matematica e Applicazioni "Renato Caccioppoli"
Scuola di dottorato: Scienze matematiche ed informatiche
Dottorato: Scienze matematiche
Ciclo di dottorato: 28
Coordinatore del Corso di dottorato:
De Giovanni,
Date: 31 March 2016
Number of Pages: 94
Uncontrolled Keywords: Partial regularity, elliptic and parabolic systems, general growth
Settori scientifico-disciplinari del MIUR: Area 01 - Scienze matematiche e informatiche > MAT/05 - Analisi matematica
Date Deposited: 08 Apr 2016 09:46
Last Modified: 07 Nov 2016 08:39


Elliptic and parabolic equations arise in the mathematical description of a wide variety of phenomena, not only in the natural science but also in engineering and economics. To mention few examples, consider problems arising in different contexts: gas dynamics, biological models, the pricing of assets in economics, composite media. The importance of these equations from the applications' point of view is equally interesting from that of analysis, since it requires the design of novel techniques to attack the always valid question of existence, uniqueness and regularity of solutions. \\ In particular, in recent years parabolic problems came more and more into the focus of mathematicians. Changing from elliptic to the parabolic case means physically to switch from the stationary to the non-stationary case, i.e. the time is introduced as an additional variable. Exactly this natural origin constitutes our interest in parabolic problems: they reflect our perception of space and time. Therefore they often can be used to model physical process, e.g. heat conduction or diffusion process. \medskip \noindent In this thesis I will principally concentrate on the regularity properties of solutions of second order systems of partial differential equations in the elliptic and parabolic context. The outline of the thesis is as follows. \smallskip \noindent After giving some preliminary results, in the 3st Chapter we consider the parabolic analogue of some regularity results already known in the elliptic setting, concerning systems becoming parabolic only in an {\it asymptotic} sense. In the standard elliptic version, these results prove the {\it Lipschitz regularity} of solutions to elliptic systems of the type $\dive a(Du)=0$, with $u: \Omega \rightarrow \R^{N}$, under the main assumption that the vector field $a: \R^{Nn}\rightarrow \R^{Nn}$ is {\it asymptotically} close, in $C^{1}$-sense, to some regular vector field $b$. Therefore, one can ask what happens when the vector field $a$ is {\it asymptotically} close, in a $C^{0}$-sense, to the regular vector field $b(\xi)=\xi$. In this direction, in the parabolic framework, the first result obtained shows that the spatial gradient of $u$ belongs to $L^{\infty}_{\rm{loc}}$. \\ The question that naturally arises is what happens in case of power $p\neq 2$, and more in general in case of general growth $\V$. \noindent Regarding the general growth $\V$, in Chapter \ref{phi}, we study variational integrals of the type \begin{equation*} \mathcal{F}(u):=\int_{\Omega} f(Du) \,dx \quad \mbox{ for } u:\Omega \rightarrow \R^{N} \end{equation*} where $\Omega$ is an open bounded set in $\R^{n}$, $n\geq 2$, $N\geq 1$. Here $f:\R^{Nn}\rightarrow \R$ is a quasiconvex continuous function satisfying a non-standard growth condition: \begin{equation*} |f(z)|\leq C(1+ \varphi(|z|) ), \quad \forall z\in \R^{Nn}, \end{equation*} where $C$ is a positive constant and $\V$ is a given $N$-function (see Section \ref{Orlicz} for more details about Orlicz functions). Exhibiting an adequate notion of strict $W^{1,\V}$-quasiconvexity at infinity, which we call $W^{1,\V}$-asymptotic quasiconvexity, we prove a partial regularity result, namely that minimizers are {\it Lipschitz} continuous on an open and dense subset of $\Omega$. \\ In the last Chapter we deal with the study of {\it local Lipschitz regularity} of weak solutions to non-linear second order parabolic systems of general growth \begin{equation}\label{Prob} u_{t}^{\beta} - \sum_{i=1}^{n} (\A_{i}^{\alpha}(Du))_{x_{i}}=0, \mbox{ in } \Omega_{T}:=\Omega \times (-T,0) \end{equation} where $\Omega$ is a bounded domain in $\R^{n}$, $n\geq 2$, $T>0$, $u:\Omega_{T} \rightarrow \R^{N}$, $N>1$ and $\A$ is a tensor having general growth, that is $\displaystyle{\A_{i}^{\alpha}(Du)= \frac{\V'(|Du|)}{|Du|} u_{x_{i}}^{\alpha}}$, where $\V$ is a given $N$-function. \\ Actually, having such result, as observed before, it is possible to prove the analogue of the first problem (studied in Chapter \ref{BMO}) in this case of nonstandard growth, considering an operator $\mathcal{A}$ that is {\it asymptotically} related to (\ref{Prob}).

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