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Estimations de Schauder et regularité höldérienne pour une classe de problèmes aux limites elliptiques singuliers. (French) Zbl 0613.35027

Sémin., Bony-Sjöstrand-Meyer Équations Dériv. Partielles 1984-1985, Exp. No. 15, 14 p. (1985).
Authors prove the Schauder estimates and Hölder continuity (in weighted Hölder spaces) of solutions to the elliptic equation with an operator \[ L=\sum^{Min(k,2m)}_{h=0}\phi^{k-h} P^{2m-h}(x,D); \] where k,m\(\in N\), \(\phi\) is a \(C^{\infty}\)-real function on \(R^ n\) with \(\Omega =\{x\in R^ n:\phi (x)>0\}\), \(\Gamma =\{x\in R^ n:\phi (x)=0\}\), \(P^{2m-h}\) are differential operators with \(C^{\infty}\) coefficients of the order less or equal to 2m-h, \(P^{2m}\) being properly elliptic on \({\bar \Omega}\). Denoting the main part of \(P^{2m- h}\) by \(P^{2m-h}_{2m-h}\) and \[ p(x,\lambda)=\sum^{Min(k,2m)}_{h=0}(-i)^{2m-h} P^{2m-h}_{2m- h}(x,\text{grad} \phi (x))\lambda (\lambda -1)...(\lambda -k+h+1), \] the authors suppose:
\(H1_{\mu}:\) for \(x\in \Gamma\) the equation \(p(x,\lambda)=0\) has no solution \(\lambda\) on the line Re\(\lambda=\mu.\)
Denote the number of all solutions of \(p(x,\lambda)=0\) with Re\(\lambda>\mu\) by \(r_{\mu}(x).\)
\(H2_{\mu}:\) \(r_{\mu}(x)\) is a constant function on \(\Gamma\) and \(\chi_{\mu}=m-h+r_{\mu}\geq 0.\)
\(H3_{\mu}:\) For \(x\in \Gamma\) and every cotangent vector \(\xi\neq 0\) the boundary value problem \[ L_ 0(x,\xi,t,D_ t)u=\sum^{Min(2m,k)}_{h=0}t^{k-h} P^{2m-h}_{2m-h}(x,\xi +\text{grad} \phi (x)D_ t)u=0;\quad B^ 0(x,\xi,D_ t)u=0 \] has only the trivial solution \(u=0\) in the space \[ C_ k^{\mu +2m- k}(R_+)=\{u\in C^{\mu +2m-k}(R_+);\quad t^ hu\in C^{\mu +2m- k+h}(R_+),\quad 0\leq h\leq k\}. \] Here \(B^ 0\) is the main part of the system of boundary operators \((B_ j)\) on \(\Gamma\) with order \(B_ j=m_ j<\mu +2m-k\). If \(\chi_{\mu}=0\), the system of boundary operators is void. Under these assumptions the following a priori estimate holds: \[ \| u\|_{C_ k^{\mu +2m-k}({\bar \Omega})}\leq C\{\| Lu\|_{C^{\mu}({\bar \Omega})}+\| B_{\mu}u\|_{\prod^{\chi -1}_{j=0}C^{2m-k+\mu -m_ j}(\Gamma)}+\| u\|_{C_ k^{\mu +2m-k-1}({\bar \Omega})}\}. \] Supposing, moreover, that there is no other solution \(p(x,\lambda)=0\) in the strip \(\mu'\leq Re\lambda \leq \mu\), we have that a solution u \(C_ k^{\mu'+2m-k}({\bar \Omega})\) with more smooth Lu and \(B_{\mu'}u\) belongs to \(C_ k^{\mu +2m-k}({\bar \Omega}).\)
The estimates are first proved on a half space. Using the Fourier transform is tangent directions the problem is reduced to an ordinary differential equation.
Reviewer: J.Stará

MSC:

35J40 Boundary value problems for higher-order elliptic equations
35B65 Smoothness and regularity of solutions to PDEs
35B45 A priori estimates in context of PDEs
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