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# A Dirichlet problem involving the divergence operator

Dr. Gyula Csato, University of Concepcion, Chile
 Speaker Dr. Gyula Csato, University of Concepcion, Chile Jan 31, 2017 from 04:00 PM to 05:00 PM LH006 vCal iCal

### Abstract

Consider the following classical and broadly treated problem: Given a function f on Ω ⊂ R^n,
find a vector field u such that

(1)  div u = f in Ω
u = 0 on ∂Ω.

It is obvious that a necessary condition is {Ω f = 0. This is also a sufficient condition. Let us generalize  the diffirential operator and introduce lower order terms. This leads to the boundary value problem

div u + ha, ui = f in Ω
u = 0 on ∂Ω,

where a is a given vector field and h , i is the scalar product. What is now the necessary and sufficient condition for solvability? What is the expected regularity result? The answer is easy, if a is of the special

form a = grad A. First I give an overview on the history and relevance of equation (??) and one method how it can be tackled. Then I present some results and conjectures about the general case. This is joint work with B. Dacorogna appearing in the following reference:

Csat ́o G. and Dacorogna B., A Dirichlet problem involving the divergence operator, Ann. Inst. H. Poincar ́e
Anal. Non Linaire, doi:10.1016/j.anihpc.2015.01.006, to appear.

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