## 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.

### MSC:

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

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