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We propose another extension of Orlicz-Sobolev spaces to metric spaces based on the concepts of the -modulus and -capacity. The resulting space is a Banach space. The relationship between and (the first extension defined in Aïssaoui (2002)) is studied. We also explore and compare different definitions of capacities and give a criterion under which is strictly smaller than the Orlicz space .
Local Lipschitz continuity of local minimizers of vectorial integrals is proved when satisfies - growth condition and is not convex. The uniform convexity and the radial structure condition with respect to the last variable are assumed only at infinity. In the proof, we use semicontinuity and relaxation results for functionals with nonstandard growth.
Theorems on the Fredholm alternative and well-posedness of the linear boundary value problem , , where and are linear bounded operators, , and , are established even in the case when is not a strongly bounded operator. The question on the dimension of the solution space of the homogeneous equation is discussed as well.
We give new criterions to decide if some vector-valued function is a local Laplace transform and apply this to the theory of local Cauchy problems. This leads to an improvement of known results and new Hille-Yosida-type theorems for local convoluted semigroups.