Regularity of solutions to non-uniformly characteristic boundary value problems for symmetric systems

Secchi

Trebeschi

2014

J Fourier Anal Appl

Section: Preliminaries and Technical Toolsmentioning

confidence: 99%

“…Of course, one has that u , u ∈ D (R n ); moreover the following relations can be easily verified (cf. [26])…”

Section: Preliminaries and Technical Toolsmentioning

confidence: 99%

On a Priori Energy Estimates for Characteristic Boundary Value Problems

Secchi

Trebeschi

2014

J Fourier Anal Appl

“…the next Section 4). We start by recalling the definition of two operators and , introduced by Nishitani and Takayama in [19], with the main property of mapping isometrically square integrable (resp. essentially bounded) functions over the half-space R n + onto square integrable (resp.…”

Section: Preliminaries and Technical Toolsmentioning

confidence: 99%

“…To prove the result of [18], the solution u to (1.1)-(1.3) is regularized by a family of tangential mollifiers J ε , 0 < ε < 1, defined by Nishitani and Takayama in [19] as a suitable combination of the operator (see Section 3) and the standard Friedrichs'mollifiers. The essential point of the analysis performed in [18] is to notice that the mollified solution J ε u solves the same problem (1.1)-(1.3), as the original solution u.…”

Section: IImentioning

confidence: 99%

See 1 more Smart Citation

Regularity of Weakly Well Posed Hyperbolic Mixed Problems With Characteristic Boundary

J. Hyper. Differential Equations

Secchi

2011

Abstract. We study the mixed initial-boundary value problem for a linear hyperbolic system with characteristic boundary of constant multiplicity. We assume the problem to be "weakly" well posed, in the sense that a unique L 2 -solution exists, for sufficiently smooth data, and obeys an a priori energy estimate with a finite loss of conormal regularity. This is the case of problems that do not satisfy the uniform Kreiss-Lopatinskiȋ condition in the hyperbolic region of the frequency domain. Under the assumption of the loss of one conormal derivative we obtain the regularity of solutions, in the natural framework of weighted anisotropic Sobolev spaces, provided the data are sufficiently smooth.

Regularity of weakly well posed non characteristic boundary value problems

J. Pseudo-Differ. Oper. Appl.

Trebeschi

2012

We consider a boundary value problem for a system of linear partial differential equations with non regular coefficients.We assume the problem to be “weakly” well posed, in the sense that a unique L2-solution exists, for sufficiently smooth data, and obeys an a priori energy estimate with a finite loss of tangential regularity with respect to the data. This is the case of problems that do not satisfy the uniform Kreiss–Lopatinskii condition in the hyperbolic region of the frequency domain. Provided the data are sufficiently smooth, we obtain the regularity of solutions, in the natural framework of weighted Sobolev spaces