We define and study variable exponent Sobolev spaces with zero boundary values. This allows us to prove that the Dirichlet energy integral has a minimizer in the variable exponent case. Our results are based on a Poincaré-type inequality, which we prove under a certain local jump condition for the variable exponent. Mathematics Subject Classifications (2000) 46E35 · 31C45 · 35J65Key words variable exponent Sobolev space · zero boundary values · Sobolev capacity · Poincaré inequality · Dirichlet energy integral

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We define Sobolev capacity on the generalized Sobolev spaceW1, p(⋅)(ℝn). It is a Choquet capacity provided that the variable exponentp:ℝn→[1,∞)is bounded away from 1 and∞. We discuss the relation between the Hausdorff dimension and the Sobolev capacity. As another application we study quasicontinuous representatives in the spaceW1, p(⋅)(ℝn).

Mathematics and statistics major master graduates of a Finnish university answered a questionnaire about the master thesis process. Their response was analyzed to find connections of the topic and supervisor selection to the factors supporting and preventing the master thesis progress. Certainty of the thesis topic and of selecting a supervisor does not seem to predict smoother thesis progress than uncertainty of the selections. Students who are not very sure but still fairly sure about the both selections provisionally meet more supporting factors and less preventing factors than other students. An interesting observation is that there are factors which support the progress of some students but prevent the progress of some other students. Mathematics teacher specialization students' experiences differ slightly from those of the other specializations students, in particular, their theses are graded lower and they hesitate more the selection of the master thesis supervisor.