{'fraction_non_alphanumeric': 0.005940594059405941, 'fraction_numerical': 0.0, 'mean_word_length': 5.325, '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}}
ADJOINT FUNCTIONS TO BOUNDARY SOLUTIONS OF DIFFERENTIAL INCLUSIONS AND SMOOTHNESS OF BARRIER SOLUTIONS ON SEMIPERMEABLE SURFACES

The problem of existence of adjoint functions to boundary solutions is considered it depends on the geometry of the attained set at the end point. This is applied to prove the smoothness of boundary solutions in the case of strictly convex right-hand side of differential inclusion which in turn permitts to show the smoothness of barrier solutions on semipermeable surfaces.