Resumen de On elliptic equations involving the 1-laplacian operator

Marta Latorre Balado

• This PhD thesis is devoted to the study of the existence, uniqueness and regularity of solutions of some elliptic equations involving the 1-Laplacian operator.

  In Chapter 1 we focus on the study of the equation - div (Du/|Du|) + g(u) |Du| = f(x) in a bounded open set U in R^N with Lipschitz boundary with the Dirichlet condition u=0 on the boundary, where the datum f denotes a nonnegative function and g is a nonnegative continuous real function. On the one hand, we deal with unbounded solutions when datum f belongs to the Marcinkiewicz space L^{N,\infty}(U), so we have to introduce the suitable concept of this kind of solutions. On the other hand, we show different results concerning existence, uniqueness or regularity of the solutions depending on the properties of function g.

  Chapter 2 is devoted to study the previous equation when we take a constant function g equal to 1 and nonnegative data f in the Lebesgue space L^1(U). We prove an existence result and a comparison principle. Moreover, we seek the optimal summability of the solution when L^q-data, with , are considered.

  Finally, in Chapter 3 we deal with an evolution problem. If L is a positive parameter, we study equation L u - div(Du/|Du|) = 0 with dynamical boundary conditions. Applying nonlinear semigroup theory we obtain a mild solution to the problem and we prove that this is, in fact, a strong solution. We also prove a comparison principle and a result which shows that the distance between the solutions depends on the distance between the data.