{'fraction_non_alphanumeric': 0.02108843537414966, 'fraction_numerical': 0.0, 'mean_word_length': 4.657692307692308, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Existence and Uniqueness of Entropy Solution for an Elliptic Problem with Nonlinear Boundary Conditions and Measure-data

In this paper, we study a class of elliptic equations on a bounded domain with nonlinear boundary conditions of type graph and measure - data. First of all, give some spaces and basic assumptions. Next, we apply the classical variational approach. So, we need an essentially bounded estimate on the solution, which is not evident to obtain directly in our problem. The obstacles which we encounter is that we cannot get rid of the non-linear term evaluated as a zero gradient and it appear at the boundary, for the part of the measure-data, a term which cannot vanish, when one uses the integration by parts formula. To overcome this difficulties, we first redefine and extend the function which appears in the third Leray-Lions-type conditions and we add a penalization term on the boundary. Secondly, we consider a smooth domain in order to work with the Sobolev spaces that are the closure of indefinitely differentiable and null functions on the bounary, and to going back later to the classical Sobolev space. Then, we assume that the domain is extensible. Next, we obtain a priori estimates and convergence results in the approach problem, which allow us to delete the penalization term. To finish, we introduce a notion of entropy solution for our main problem and prove that it is the limit of the solution obtained in the variational case.