Definition and existence of renormalized solutions of elliptic equations with general measure data

A note on a critical problem with natural growth in the gradient

“…Finally, by (9) and using the same ideas as in [2] (see also [4]), we conclude that u satisfies the equation in the renormalized sense (see [11]). …”

“…For every δ ∈ (0, 1) and k > 0, defining E n,k ≡ E ∩ {v n ≤ k}, we obtain The last integral is uniformly bounded with respect to n, therefore the corresponding term can be made small by choosing k large enough. Moreover, for every k > 0, T k (w n ) → T k (w) strongly in D 1,2 0,(N−2)/2 ( ) (see [11]), so the integral E |∇T k (w n )| 2 |x| −(N−2) dx is uniformly small if the measure of E is small enough. The equi-integrability of the sequence |∇v n | 2 |x| −(N−2) follows immediately.…”

Existence of entropy solutions for nonlinear elliptic degenerate anisotropic equations

“…; /, and properties (9) and (10) of Definition 2.2 are satisfied. Thus, u is an entropy solution to the Dirichlet problem (7), (8). The theorem is proved.…”

“…As is generally known, an effective approach to the solvability of second-order equations in divergence form with L 1 -right-hand sides has been proposed in [1]. In this connection we also mention a series of other close investigations for nondegenerate isotropic nonlinear second-order equations with L 1 -data and measures, entropy and renormalized solutions [2][3][4][5][6][7][8][9][10].…”

Measure-Valued Solutions to a Harvesting Game with Several Players

“…On the other hand, Theorem 3 cannot have a direct counterpart valid in dimension n ≥ 2. Indeed, the results in [10,11] indicate that the multidimensional multi-dimensional optimization problem is not well posed. To In order to have the existence of an optimal solution, an additional cost term is needed.…”