We are interested in algorithms for constructing surfaces Γ of possibly small measure that separate a given domain Ω into two regions of equal measure. Using the integral formula for the total gradient variation, we show that such separators can be constructed approximatively by means of sign changing eigenfunctions of the p-Laplacians, p → 1, under homogeneous Neumann boundary conditions. These eigenfunctions turn out to be limits of steepest descent methods applied to suitable norm quotients.
We study extension of p-trigonometric functions sinp and cosp to complex domain. For p = 4, 6, 8, . . ., the function sinp satisfies the initial value problem which is equivalent to (∗) −(u ′ ) p−2 u ′′ − u p−1 = 0, u(0) = 0, u ′ (0) = 1 in R. In our recent paper, Girg, Kotrla (2014), we showed that sinp(x) is a real analytic function for p = 4, 6, 8, . . . on (−πp/2, πp/2), where πp/2 = R 1 0 (1 − s p ) −1/p. This allows us to extend sinp to complex domain by its Maclaurin series convergent on the disc {z ∈ C: |z| < πp/2}. The question is whether this extensions sinp(z) satisfies (∗) in the sense of differential equations in complex domain. This interesting question was posed by Došlý and we show that the answer is affirmative. We also discuss the difficulties concerning the extension of sinp to complex domain for p = 3, 5, 7, . . . Moreover, we show that the structure of the complex valued initial value problem (∗) does not allow entire solutions for any p ∈ ℕ, p > 2. Finally, we provide some graphs of real and imaginary parts of sinp(z) and suggest some new conjectures.
We study the Dirichlet boundary value problem for the p-Laplacian of the form −∆pu − λ1|u| p−2u = f in Ω, u = 0 on ∂Ω, where Ω ⊂ N is a bounded domain with smooth boundary ∂Ω, N ≥ 1, p > 1, f ∈ C(Ω) and λ1 > 0 is the first eigenvalue of ∆p. We study the geometry of the energy functional Ep(u) = 1⁄ p ∫ Ω |∇u| p − λ1⁄ p ∫ Ω |u| p − ∫ Ω fu and show the difference between the case 1 <p< 2 and the case p > 2. We also give the characterization of the right hand sides f for which the above Dirichlet problem is solvable and has multiple solutions.
We investigate two boundary value problems for the second order differential equation with p-Laplacian (a(t)Φp(x ′ ))′ = b(t)F(x), t ∈ I = [0, ∞), where a, b are continuous positive functions on I. We give necessary and sufficient conditions which guarantee the existence of a unique (or at least one) positive solution, satisfying one of the following two boundary conditions: i) x(0) = c > 0, lim t→∞ x(t) = 0; ii) x ′ (0) = d < 0, lim t→∞ x(t) = 0.
In the paper the differential inequality ∆pu + B(x, u) ≤ 0, where ∆pu = div(||∇u|| p−2∇u), p > 1, B(x, u) ∈ C(Rn × R, R) is studied. Sufficient conditions on the function B(x, u) are established, which guarantee nonexistence of an eventually positive solution. The generalized Riccati transformation is the main tool.
In this paper I discuss two questions on p-Laplacian type operators: I characterize sets that are removable for Hölder continuous solutions and then discuss the problem of existence and uniqueness of solutions to−div(|∇u| p−2∇u) = µ with zero boundary values; here µ is a Radon measure. The joining link between the problems is the use of equations involving measures.
We study the quasilinear elliptic problem with multivalued terms.We consider the Dirichlet problem with a multivalued term appearing in the equation and a problem of Neumann type with a multivalued term appearing in the boundary condition. Our approach is based on Szulkin’s critical point theory for lower semicontinuous energy functionals.
The Picone-type identity for the half-linear second order partial differential equation n∑ i=1 ∂ ⁄ ∂xi Φ (∂u ⁄ ∂xi) + c(x)Φ(u) = 0, Φ(u) := |u| p−2 u, p > 1, is established and some applications of this identity are suggested.
General mathematical theories usually originate from the investigation of particular problems and notions which could not be handled by available tools and methods. The Fučík spectrum and the p-Laplacian are typical examples in the field of nonlinear analysis. The systematic study of these notions during the last four decades led to several interesting and surprising results and revealed deep relationship between the linear and the nonlinear structures. This paper does not provide a complete survey. We focus on some pioneering works and present some contributions of the author. From this point of view the list of references is by no means exhaustive.