Papers by Yannakakis Nikos
In this paper we study a Neumann problem with non-homogeneous boundary condition, where the p(x)-... more In this paper we study a Neumann problem with non-homogeneous boundary condition, where the p(x)-Laplacian is involved. In particular we assume that p(·) is a step function defined in a domain Ω and equals to 1 in a subdomain Ω 1 and 2 in its complementary Ω 2. By considering a suitable sequence p k of variable exponents such that p k → p and replacing p with p k in the original problem, we prove the existence of a solution u k for each of those intermediate ones. We also show, that under a hypotheses concerning the boundary data g, the limit of the sequence (u k) is a function u, which belongs to the space of functions of bounded variation and is a solution to the original p(·)-problem.
In this paper we study a Neumann problem with non-homogeneous boundary condition, where the p(x)-... more In this paper we study a Neumann problem with non-homogeneous boundary condition, where the p(x)-Laplacian is involved and p = ∞ in a subdomain. By considering a suitable sequence p k of bounded variable exponents such that p k → p and replacing p with p k in the original problem, we prove the existence of a solution u k for each of those intermediate ones. We show that the limit of the u k exists and after giving a variational characterization of it, in the part of the domain where p is bounded, we show that it is a viscosity solution in the part where p = ∞. Finally, we formulate the problem of which this limit function is a solution in the viscosity sense.
Thesis Chapters by Yannakakis Nikos
Most of the times, in problems where the p(x)-Laplacian is involved, the variable exponent p(·) i... more Most of the times, in problems where the p(x)-Laplacian is involved, the variable exponent p(·) is assumed to be bounded. The main reason for this is to be able to apply standard variational methods. The aim of this paper is to present the work that has been done so far, in problems where the variable exponent p(·) equals infinity in some part of the domain. In this case the infinity Laplace operator arises naturally and the notion of weak solution does not apply in the part where p(·) becomes infinite. Thus the notion of viscosity solution enters in to the picture. We study both the Dirichlet and the Neumann case.
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Papers by Yannakakis Nikos
Thesis Chapters by Yannakakis Nikos