On linear degenerate elliptic PDE systems with constant coefficients

Katzourakis, N. (2016) On linear degenerate elliptic PDE systems with constant coefficients. Advances in Calculus of Variations, 9 (3). pp. 283-291. ISSN 1864-8266 doi: 10.1515/acv-2015-0004

Abstract/Summary

Let A be a symmetric convex quadratic form on RNn and Ω ⊂ Rn a bounded convex domain. We consider the problem of existence of solutions u: Ω ⊂ Rn → RN to the problem ⎧⎩⎨⎪⎪∑β=1N∑i,j=1nAαiβjD2ijuβu=fα=0inΩ,on∂Ω, when f∈L2(Ω,RN). Problem (1) is degenerate elliptic and it has not been considered before without the assumption of strict rank-one convexity. In general, it may not have even distributional solutions. By introducing an extension of distributions adapted to (1), we prove existence, partial regularity and by imposing an extra condition uniqueness as well. The satisfaction of the boundary condition is also an issue due to the low regularity of the solution. The motivation to study (1) and the method of the proof arose from recent work of the author [10] on generalised solutions for fully nonlinear systems.