Abstract
It is known (cf. Martell et al. in Rev Mat Iberoam 32(3):913–970, 2016) that the \(L^p\)-Dirichlet problem for constant coefficient second-order systems satisfying the Legendre-Hadamard (strong) ellipticity condition is well posed in the upper half-space. However, this may fail if only weak ellipticity for the system in question is assumed. In this paper we shall show that the aforementioned failure is at a fundamental level, in the sense that there exist systems which are weakly elliptic (i.e., their characteristic matrix is invertible) for which the \(L^p\)-Dirichlet problem in the upper half-space is not even Fredholm solvable.
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Dedicated to Vladimir Maz’ya with great admiration and high esteem.
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The authors gratefully acknowledge partial support from the Simons Foundation (through Grants \(\#\,\)426669, \(\#\,\)616050, \(\#\,\)637481), as well as NSF (Grant \(\#\,\)1900938)
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Mitrea, D., Mitrea, I. & Mitrea, M. Failure of Fredholm solvability for the Dirichlet problem corresponding to weakly elliptic systems. Anal.Math.Phys. 11, 85 (2021). https://doi.org/10.1007/s13324-021-00521-4
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DOI: https://doi.org/10.1007/s13324-021-00521-4
Keywords
- Bitsadze’s operator
- Weakly elliptic system
- Legendre–Hadamard ellipticity
- Dirichlet problem
- Upper half-space
- Space of admissible boundary data
- Fredholm solvability
- Dirichlet-to-Neumann map
- Riesz transform
- Poisson kernel