On boundary values of solutions of a quasi-linear partial differential equation of elliptic type
Consider the quasi-linear elliptic equation \[ (1)\quad - \sum^{n}_{i,j=1}D_ i(a_{ij}(x,u)D_ ju)+b(x,u,Du)+\lambda u=0 \] in a bounded domain Q of \({\mathbb{R}}_ n\) with \(C^ 2\) boundary \(\partial Q\). In addition to some standard hypotheses, it is assumed that \[ K^{- 1}| \xi |^ 2\leq \sum^{n}_{i,j=1}a_{ij}(x,u)\xi_ i\xi_ j\leq K| \xi |^ 2,\quad | b(x,u,s)| \leq f(x)+C(| u| +| s|), \] \[ (2)\quad \int_{Q}f(x)^ 2 r(x)^{\theta} dx<\infty \text{ for some } \theta <3, \] where \(r(x)=dist(x,\partial Q)\) and K and C are positive constants. Thus, the ellipticity hypothesis and the growth conditions in u and Du are very strong, but the hypothesis (2) on f is rather weak and nonstandard. In Theorem 1 it is proved that if \(u\in W^{1,2}_{loc}(Q)\) is a solution of (1) such that (3) \(\int_{Q}| Du(x)^ 2 r(x)dx<\infty\), then u has an \(L^ 2\) trace on \(\partial Q\). Notice that for a function u (which may not be a solution of the equation) \(\int_{Q}| Du(x)|^ 2 r(x)^{1-\epsilon} dx<\infty\) implies that u has an \(L^ 2\) trace on \(\partial Q\) for any \(\epsilon >0\), but not for \(\epsilon =0\) [see \textit{A. Kufner}, ''Weighted Sobolev spaces'' (1985; Zbl 0567.46009), p. 75]. In Theorem 2 an energy estimate involving the integral (3) is proved (for \(\lambda\) large enough). In Theorem 3 it is proved that for \(\lambda\) large enough and for any \(\phi\in L^ 2(\partial Q)\) there exists a solution \(u\in W^{1,2}_{loc}(Q)\) of Equation (1) (with b(x,u,Du) replaced by c(x,u)u-\(f(x))\) such that \(u=\phi\) on \(\partial Q\). Notice that it is not assumed that \(\phi \in H^{1/2}(\partial Q).\) The methods of this paper are based partially by the author and \textit{B. Thompson} [Ann. Pol. Math. 42, 45-71 (1983; Zbl 0564.35032)], and the author [Manuscr. Math. 40, 91-108 (1982; Zbl 0503.35036)]. A key idea is the use of a test function \(\nu\) defined by \(\nu (x)=u(x)\cdot (\rho (x)- \delta)\) if \(r(x)>\delta\); \(\nu (x)>0\) if r(x)\(\leq \delta\), where \(\rho\) (x) is a ''smooth \(C^ 2\) modification'' of r(x). The construction and use of \(\rho\) (x) is based on \textit{D. Gilbarg} and \textit{N. S. Trudinger} [''Elliptic partial differential equations of second order'', 2nd ed. (1983; Zbl 0562.35001), p. 355] and on \textit{V. P. Mikhajlov} [Mat. Sb., Nov. Ser. 101(143), 163-188 (1976; Zbl 0345.35033)].
scientific article; zbMATH DE number 4008931
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