Entropy solutions of the Dirichlet problem for a class of non-linear elliptic fourth-order equations with right-hand sides in $ L^1$

Abstract

For partial differential equations written in conservative form a remarkable link between potential symmetries and direct reduction methods of order two is enlightened.

On the improvement of summability of generalized solutions of the Dirichlet problem for nonlinear equations of the fourth order with strengthened ellipticity

“…In these papers, on the basis of the development of the approach proposed in [6], results on the existence and properties of summability of entropy solutions and W-solutions (which are weaker than generalized solutions) were obtained for the corresponding Dirichlet problem. In the case where the right-hand sides of equations belong to spaces close to L 1 (i.e., to spaces L m with m greater than 1 and smaller than a certain m 0 ), the properties of summability of entropy solutions and W-solutions of the same problem as in [3,4] and in the present paper were described in [7]. A further increase in the summability of the right-hand sides of the considered equations in the scale of Lebesgue spaces leads to generalized solutions.…”

“…2 in[4], using properties (42) -(44) we establish that h k ( u ) ∈ ) for any n-dimensional multiindex α, α = 1, one has D h u k α ( ) = ′ h u D u k ( ) α almost everywhere on Ω;…”

On the convergence of functions from a Sobolev space satisfying special integral estimates

“…The direct definition of the functions δ i u for u ∈ • T 1,p (Ω) by equality (3) and the proof of Proposition 1 can be found in [2]. Lemma 1 is, in fact, proved in [3]. For similar results with more exact estimates used instead of (5), see [4,5].…”

“…Thus, in particular, this is true for higher-order equations whose coefficients satisfy the condition of strong ellipticity (Skrypnik condition [10]). The problems of existence of solutions of these equations with L 1 -right-hand sides are studied in [3,11].…”

On Conditions of the Nonexistence of Solutions of Nonlinear Equations with Data from Classes Close to <em>L</em>¹

“…Some results on the existence of entropy and weak solutions of the Dirichlet problem for nonlinear elliptic high-order equations with coefficients satisfying a strengthened coercivity condition and L 1 -data were obtained for instance in [4] and [5]. In this connection see also [2,Chapter 2] where a theory of the existence and properties of entropy and weak solutions of the Dirichlet problem for nonlinear fourth-order equations with strengthened coercivity and data from L 1 and classes close to L 1 is presented.…”