×

On the boundedness of the first derivatives of the solution of the Dirichlet problem in a domain with smooth nonregular boundary. (Russian) Zbl 0177.14403

Let \(D\) be a bounded domain and suppose that \(\partial D\) is of class \(C^1\) and has bounded total angular variation. Suppose also that the curvature of \(\partial D\) is continuous except possibly at a point \(0\). Let \(F\subset \partial D\) be closed; let \(\ell\) denote the length along \(\partial D\) measured from \(0\) and let \(\ell_F\) be the length of \(F\). Suppose that \(\tau_F^{-}\) is the negative variation of the total angular variation over \(F\). Let
\[ \alpha(\ell) = \sup\{\tau_F^{-} \mid F\subset (\vert z-0\vert< \varepsilon,\ \ell_F\le1)\}.\]
Then: Theorem. Let \(au_{xx}+ 2bu_{xy}+ cu_{yy} = f\) be uniformly elliptic and suppose that \(a,b,c\) are bounded and measurable. If \(\partial D\) satisfies \(\int_0 \alpha(\ell)\frac{d\ell}{\ell}<\infty\) and if \(f\in L_p(D)\), \(p\ge 2\), then any solution \(u\in \overset{\circ}{W_2^2}(D)\) of the equation satisfies \(\operatorname{ess\ sup} \vert\nabla u\vert\le C\, \Vert f\Vert_{L_p}(D)\).
Reviewer: N. du Plessis

MSC:

35J40 Boundary value problems for higher-order elliptic equations
PDFBibTeX XMLCite