Estimates for solutions of a parameter-elliptic multi-order system of differential equations. (English) Zbl 1202.35077
The authors consider an elliptic system of partial differential equations with spectral parameter \(\lambda\):
\[ \begin{aligned} A(x,D) u(x)-\lambda u(x)= f(x)&\quad\text{in }\Omega,\\ B_j(x, D)u(x)= g_j(x)&\quad\text{on }\Gamma\quad\text{for }j= 1,\dots, M, \end{aligned} \]
where \(\Omega\) is a bounded set in \(\mathbb{R}^n\) with boundary \(\Gamma\). The notations are vectorial, in particular \(A(x,D)\) is a square matrix of operators \(A_{jk}(x,D)\) of different orers. Precise a-priori estimates are proved, under limited smoothness assumptions, generalizing existing results, cf. R. Denk and L. Volevich [Providence, RI: American Mathematical Society (AMS). Transl., Ser. 2, Am. Math. Soc. 206, 29–64 (2002; Zbl 1056.35058)].
\[ \begin{aligned} A(x,D) u(x)-\lambda u(x)= f(x)&\quad\text{in }\Omega,\\ B_j(x, D)u(x)= g_j(x)&\quad\text{on }\Gamma\quad\text{for }j= 1,\dots, M, \end{aligned} \]
where \(\Omega\) is a bounded set in \(\mathbb{R}^n\) with boundary \(\Gamma\). The notations are vectorial, in particular \(A(x,D)\) is a square matrix of operators \(A_{jk}(x,D)\) of different orers. Precise a-priori estimates are proved, under limited smoothness assumptions, generalizing existing results, cf. R. Denk and L. Volevich [Providence, RI: American Mathematical Society (AMS). Transl., Ser. 2, Am. Math. Soc. 206, 29–64 (2002; Zbl 1056.35058)].
Reviewer: Luigi Rodino (Torino)
MSC:
| 35J46 | First-order elliptic systems |
| 35J47 | Second-order elliptic systems |
| 35J48 | Higher-order elliptic systems |
| 35S15 | Boundary value problems for PDEs with pseudodifferential operators |
| 35B45 | A priori estimates in context of PDEs |
Citations:
Zbl 1056.35058References:
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