Optimal \(L^{p}\)- \(L^{q}\)-estimates for parabolic boundary value problems with inhomogeneous data. (English) Zbl 1210.35066

Summary: We investigate vector-valued parabolic initial boundary value problems \({(\mathcal A(t,x,D)}\), \({\mathcal B_j(t,x,D))}\) subject to general boundary conditions in domains \(G\) in \({\mathbb R^n}\) with compact \(C^{2 m}\)-boundary. The top-order coefficients of \({\mathcal A}\) are assumed to be continuous. We characterize optimal \(L^{p}\)- \(L^{q}\)-regularity for the solution of such problems in terms of the data. We also prove that the normal ellipticity condition on \({\mathcal A}\) and the Lopatinskii-Shapiro condition on \({(\mathcal A, \mathcal B_1,\dots, \mathcal B_m)}\) are necessary for these \(L^{p}\)- \(L^{q}\)-estimates. As a byproduct of the techniques being introduced we obtain new trace and extension results for Sobolev spaces of mixed order and a characterization of Triebel-Lizorkin spaces by boundary data.


35J40 Boundary value problems for higher-order elliptic equations
42B15 Multipliers for harmonic analysis in several variables
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