Nonhomogeneous boundary value problems in Orlicz–Sobolev spaces

We study the nonlinear Dirichlet problem −div(log(1+|∇u|q)|∇u|p−2∇u) = −λ|u|p−2u+|u|r−2u in Ω, u = 0 on ∂Ω, where Ω is a bounded domain in RN with smooth boundary, while p, q and r are real numbers satisfying p,q> 1, p + q < min{N,r}, r < (Np − N + p)/(N − p). The main result of this Note establishes that for any λ> 0 this boundary value problem has infinitely many solutions in the Orlicz–Sobolev space W10 LΦ(Ω), where Φ(t) = ∫ t 0 log(1 + |s|q) · |s|p−2s ds

Nonhomogeneous boundary value problems in anisotropic Sobolev spaces

We study the nonlinear boundary value problem −∑Ni=1(|uxi |pi(x)−2uxi)xi = λ|u|q(x)−2u in Ω, u = 0 on ∂Ω, where Ω ⊂RN (N 3) is a bounded domain with smooth boundary, λ is a positive real number, and the continuous functions pi and q satisfy 2 pi(x) < N and q(x)> 1 for any x ∈ Ω and any i ∈ {1,...,N}. By analyzing the growth of the functions pi and q we prove in this Note several existence results in Sobolev spaces with variable exponents

A New Approach to Solve Non-Fourier Heat Equation via Empirical Methods Combined with the Integral Transform Technique in Finite Domains

This chapter deals with the validity/limits of the integral transform technique on finite domains. The integral transform technique based upon eigenvalues and eigenfunctions can serve as an appropriate tool for solving the Fourier heat equation, in the case of both laser and electron beam processing. The crux of the method consists in the fact that the solutions by mentioned technique demonstrate strong convergence after the 10 eigenvalues iterations, only. Nevertheless, the method meets with difficulties to extend to the case of non-Fourier equations. A solution is however possible, but it is bulky with a weak convergence and requires the use of extra-boundary conditions. To surpass this difficulty, a new mix approach is proposed with this chapter resorting to experimental data, in order to support a more appropriate solution. The proposed method opens in our opinion a beneficial prospective for either laser or electron beam processing

The Monotonicity of the Principal Frequency of the Anisotropic pp-Laplacian

Let $D>1$ be a fixed integer. Given a smooth bounded, convex domain $\Omega \subset \mathbb{R}^D$ and $H:\mathbb{R}^D\rightarrow [0,\infty )$ a convex, even, and $1$-homogeneous function of class $C^{3,\alpha }(\mathbb{R}^D\setminus \lbrace 0\rbrace )$ for which the Hessian matrix $D^2(H^p)$ is positive definite in $\mathbb{R}^D\setminus \lbrace 0\rbrace$ for any $p\in (1,\infty )$, we study the monotonicity of the principal frequency of the anisotropic $p$-Laplacian (constructed using the function $H$) on $\Omega$ with respect to $p\in (1,\infty )$. As an application, we find a new variational characterization for the principal frequency on domains $\Omega$ having a sufficiently small inradius. In the particular case where $H$ is the Euclidean norm in $\mathbb{R}^D$, we recover some recent results obtained by the first two authors in [3, 4]