$W^{2,p}$-A~priori estimates for the neutral Poincar\'e problem

zbMATH Open1159.35328arXiv1110.2469MaRDI QIDQ3425555

Publication date: 26 February 2007

Abstract: A degenerate oblique derivative problem is studied for uniformly elliptic operators with low regular coefficients in the framework of Sobolev's classes

W^{2, p} (O m e g a)

for {em arbitrary}

p > 1 .

The boundary operator is prescribed in terms of a directional derivative with respect to the vector field

l

that becomes tangential to

p a r t i a l O m e g a

at the points of some non-empty subset

E s u b s e t p a r t i a l O m e g a

and is directed outwards

O m e g a

on

p a r t i a l O m e g a s e t m i n u s E .

Under quite general assumptions of the behaviour of

l,

we derive {it a priori} estimates for the

W^{2, p} (O m e g a)

-strong solutions for any

p i n (1, i n f t y) .

Full work available at URL: https://arxiv.org/abs/1110.2469

zbMATH Keywords

a priori estimates Poincaré problem strong solution $L^p$-Sobolev spaces uniformly elliptic operator neutral vector field

Mathematics Subject Classification ID

Boundary value problems for second-order elliptic equations (35J25) Ill-posed problems for PDEs (35R25) A priori estimates in context of PDEs (35B45) PDEs with low regular coefficients and/or low regular data (35R05) Subelliptic equations (35H20)

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