In this paper, under very general assumptions, we prove existence and regularity of distributional solutions to homogeneous Dirichlet problems of the formis a continuous function which may become singular at s = 0 + , and f is a nonnegative datum in L N,∞ (Ω) with suitable small norm. Uniqueness of solutions is also shown provided h is decreasing and f > 0. As a by-product of our method a general theory for the same problem involving the p-laplacian as principal part, which is missed in the literature, is established. The main assumptions we use are also further discussed in order to show their optimality.
Abstract. We prove a uniqueness result for BV solutions of scalar conservation laws with discontinuous flux in several space dimensions. The proof is based on the notion of kinetic solution and on a careful analysis of the entropy dissipation along the discontinuities of the flux.
This paper is concerned with the Dirichlet problem for an equation involving the 1–Laplacian operator normalΔ1u:=divfalse(Dufalse|Dufalse|false) and having a singular term of the type ffalse(xfalse)uγ. Here f∈LNfalse(normalΩfalse) is nonnegative, 0<γ⩽1 and normalΩ is an open bounded set with Lipschitz‐continuous boundary. We prove an existence result for a concept of solution conveniently defined. The solution is obtained as limit of solutions to p‐Laplacian type problems. Moreover, when f(x)>0 almost everywhere, the solution satisfies those features that might be expected as well as a uniqueness result. We also give explicit one–dimensional examples that show that, in general, uniqueness does not hold. We remark that the Anzellotti theory of L∞–divergence–measure vector fields must be extended to deal with this equation.
A chain rule in the space L 1 (div; Ω) is obtained under weak regularity conditions. This chain rule has important applications in the study of lower semicontinuity problems for general functionals of the form Ω f (x, u, ∇u) dx with respect to strong convergence in L 1 (Ω) . Classical results of Serrin and of De Giorgi, Buttazzo and Dal Maso are extended and generalized.
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