The $L^{p}$ regularity problem on Lipschitz domains

Abstract We consider the maximal regularity problem for a PDE of linear acoustics, named the Van Wijngaarden–Eringen equation, that models the propagation of linear acoustic waves in isothermal bubbly liquids, wherein the bubbles are of uniform radius. If the dimensionless bubble radius is greater than one, we prove that the inhomogeneous version of the Van Wijngaarden–Eringen equation, in a cylindrical domain, admits maximal regularity in Lebesgue spaces. Our methods are based on the theory of operator-valued Fourier multipliers.

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On the explicit representation of the trace space $$H^{\frac{3}{2}}$$ and of the solutions to biharmonic Dirichlet problems on Lipschitz domains via multi-parameter Steklov problems

Revista Matemática Complutense ◽

10.1007/s13163-021-00385-z ◽

2021 ◽

Author(s):

Pier Domenico Lamberti ◽

Luigi Provenzano

Keyword(s):

Fourier Series ◽

Dirichlet Problem ◽

Lipschitz Domain ◽

Spectral Properties ◽

Explicit Representation ◽

Lipschitz Domains ◽

Dirichlet Problems ◽

Trace Space ◽

In Series ◽

Definition Of

AbstractWe consider the problem of describing the traces of functions in $$H^2(\Omega )$$ H 2 ( Ω ) on the boundary of a Lipschitz domain $$\Omega $$ Ω of $$\mathbb R^N$$ R N , $$N\ge 2$$ N ≥ 2 . We provide a definition of those spaces, in particular of $$H^{\frac{3}{2}}(\partial \Omega )$$ H 3 2 ( ∂ Ω ) , by means of Fourier series associated with the eigenfunctions of new multi-parameter biharmonic Steklov problems which we introduce with this specific purpose. These definitions coincide with the classical ones when the domain is smooth. Our spaces allow to represent in series the solutions to the biharmonic Dirichlet problem. Moreover, a few spectral properties of the multi-parameter biharmonic Steklov problems are considered, as well as explicit examples. Our approach is similar to that developed by G. Auchmuty for the space $$H^1(\Omega )$$ H 1 ( Ω ) , based on the classical second order Steklov problem.

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On Sub Regular OL Forms

Fundamenta Informaticae ◽

10.3233/fi-1981-4107 ◽

1981 ◽

Vol 4 (1) ◽

pp. 135-149

Author(s):

J. Albert ◽

H.A. Maurer ◽

Th. Ottmann

Keyword(s):

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Regular Languages ◽

Regularity Problem ◽

Necessary And Sufficient

We present necessary and sufficient conditions for an OL form F to generate regular languages only. The conditions at issue can be effectively checked, whence the “regularity problem for OL forms” is proven decidable.

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