On some dissipative boundary value problems for the Laplacian

Fujiwara, Daisuke; Uchiyama, Koichi

doi:10.2969/jmsj/02340625

Cited by 29 publications

(6 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We know that P t is a positive elliptic pseudo-differential operator of order 1 on da*. See, Fujiwara-Uchiyama [5], Inoue [12], Visik [25]. Using this operator, we reduce (3.10) to the problem…”

Section: J(0mentioning

confidence: 99%

See 1 more Smart Citation

On a Mixed Problem for d'Alembertian with a Mixed Boundary Condition — An Example of Moving Boundary

Inoue

1975

Publ. Res. Inst. Math. Sci.

View full text Add to dashboard Cite

Section: J(0mentioning

confidence: 99%

“…We denote by B E the boundary operator defined by 5) In Lemma 3.1 of Ikawa [8], he puts w m+1 =0 without mention it clearly.…”

Section: Proposition 56 Let K Be a Non-negative Integer And Cp(x) Bmentioning

confidence: 99%

On a Mixed Problem for d'Alembertian with a Mixed Boundary Condition — An Example of Moving Boundary

Inoue

1975

Publ. Res. Inst. Math. Sci.

View full text Add to dashboard Cite

“…Then it follows from [27, Section 10.7] (cf. [6]) that the complete symbol t(x ′ , ξ ′ ) of the pseudo-differential operator T 1 = a(x ′ )Π 1 + b(x ′ ) is given by the formula…”

Section: Formulation Of the Homogeneous Robin Problemmentioning

confidence: 99%

Spectral analysis of the hypoelliptic Robin problem

Taira

2018

Ann Univ Ferrara

View full text Add to dashboard Cite

This paper is devoted to a functional analytic approach to the study of the hypoelliptic Robin problem for a second-order, uniformly elliptic differential operator with a complex parameter λ, under the probabilistic condition that either the absorption phenomenon or the reflection phenomenon occurs at each point of the boundary. We solve the long-standing open problem of the asymptotic eigenvalue distribution for the homogeneous Robin problem when |λ| tends to ∞. More precisely, we prove the spectral properties of the closed realization of the uniformly elliptic differential operator, similar to the elliptic (non-degenerate) case. However, in the degenerate case we cannot use Green's formula to characterize the adjoint operator of the closed realization. Hence, we shift our attention to its resolvent. In the proof, we make use of the Boutet de Monvel calculus in order to study the resolvent and its adjoint in the framework of L 2 Sobolev spaces.

show abstract

“…In this case, we can write down the complete symbol p(x , ξ ) of Π as follows (cf. [16], [38,Section 10.7]):…”

Section: 2])mentioning

confidence: 99%

Analytic semigroups for the subelliptic oblique derivative problem

Taira

2017

J. Math. Soc. Japan

View full text Add to dashboard Cite

This paper is devoted to a functional analytic approach to the subelliptic oblique derivative problem for second-order, elliptic differential operators with a complex parameter λ. We prove an existence and uniqueness theorem of the homogeneous oblique derivative problem in the framework of L p Sobolev spaces when |λ| tends to ∞. As an application of the main theorem, we prove generation theorems of analytic semigroups for this subelliptic oblique derivative problem in the L p topology and in the topology of uniform convergence. Moreover, we solve the long-standing open problem of the asymptotic eigenvalue distribution for the subelliptic oblique derivative problem. In this paper we make use of Agmon's technique of treating a spectral parameter λ as a second-order elliptic differential operator of an extra variable on the unit circle and relating the old problem to a new one with the additional variable.

show abstract

On some dissipative boundary value problems for the Laplacian

Cited by 29 publications

References 7 publications

On a Mixed Problem for d'Alembertian with a Mixed Boundary Condition — An Example of Moving Boundary

On a Mixed Problem for d'Alembertian with a Mixed Boundary Condition — An Example of Moving Boundary

Spectral analysis of the hypoelliptic Robin problem

Analytic semigroups for the subelliptic oblique derivative problem

Contact Info

Product

Resources

About