Calculus of Variations and Geometric Measure Theory
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L. Greco - C. Sbordone

Sharp Upper Bounds for the Degree of Regularity of the Solutions to an Elliptic Equation

created on 08 Jan 2002
modified on 18 Dec 2002


Published Paper

Inserted: 8 jan 2002
Last Updated: 18 dec 2002

Journal: Communications in PDE
Volume: 27
Number: 5\&6
Pages: 945-952
Year: 2002


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\vglue 2truein {\parindent 0pt \parfillskip 0pt plus1fil \obeylines \everypar{\hfil}{\baselineskip 1.2 \baselineskip % \bf SHARP UPPER BOUNDS FOR THE DEGREE OF REHULARITY OF THE SOLUTIONS TO AN ELLIPTIC EQUATION} \noindent Luigi Greco\qquad Carlo Sbordone \noindent Dipartimento di Matematica e Applicazioni ``R. Caccioppoli" Università degli Studi, Via Cintia -- 80126 NAPOLI, ITALY {\sl email:}\\qquad \noindent \noindent }

\section{ABSTRACT} It is well known that local solutions to the linear equation $$\diverg A(x)\nabla u=0\eqno(1.1)$$ in an open set $\Omega\subseteq \realnumber^n$, are locally Hölder continuous. Here we assume $\vphantom{A}^tA=A$ satisfying the uniform ellipticity condition $${
2\over K}\le \langle A(x)\xi\,,\,\xi\rangle\le K\,
2\,,\eqno(1.2)$$ with $K\ge 1$, for a.e.\ $x\in \Omega$ and all $\xi\in \realnumber^n$.\par In dimension $n=2$ the best Hölder exponent is known to be $1/K$. The optimality of such an exponent is seen by considering the functions $$uj(x)=xj\,
{{1\over K}-1},\qquad j=1,2\eqno(1.3)$$ and the coefficient matrix $$A(x)={1\over K}\,I+\left(K-{1\over K}\right)\,{x\otimes x\over
2}\,.\eqno(1.4)$$ If $K>1$, the functions $u_j$ defined in (1.3) belong to $C^{0,{1\over K}}_{\fam 0 loc}$, but not to $C^{0,\beta}_{\fam 0 loc}$ for any $\beta>1/K$.\par In any dimension, equation (1.1) with the coefficient matrix (1.4) has the solutions $$uj(x)=xj\,
{\alpha-1},\qquad j=1,2,\ldots,n\,,\eqno(1.5)$$ where $$\alpha={1\over 2}\sqrt{(n-2)2+4\,{n-1\over K2}}-{n-2\over 2}\eqno(1.6)$$ (see Lemma 1 below). Notice that for $n=2$ formula (1.6) reduces to the equality $\alpha=1/K$ and (1.5) reduce to (1.3). As above, the functions defined in (1.5) belong to $C^{0,\alpha}_{\fam 0 loc} (\Omega)$, but not to $C^{0,\beta}_{\fam 0 loc}(\Omega)$ for any $\beta>\alpha$.\par Our aim here is to show that, modulo a nonzero moment condition, all solutions of (1.1), (1.4) enjoy this same property. That is why we call the functions in (1.3)-(1.5) {\sl principal solutions} of (1.1). In other words, the principal solutions govern the regularity properties of other solutions. We also study higher order derivatives of the solutions, by means of higher order moments.\end

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