Convergence to equilibria of solutions to a conserved Phase-Field system with memory
Sergiu Aizicovici Hana Petzeltová
We show that the trajectories of a conserved phase-field model with memory are compact in the space of continuous functions and, for an exponential relaxation kernel, we establish the convergence of solutions to a single stationary state as time goes to infinity. In the latter case, we also estimate the rate of decay to equilibrium.
keywords: Conserved phase-field systems; memory effects; convergence to equilibria.
Anti-periodic solutions to a class of non-monotone evolution equations
Sergiu Aizicovici Simeon Reich
We establish the existence of solutions to an anti-periodic non-monotone boundary value problem. Our approach relies on a combination of monotonicity and compactness methods.
keywords: Anti-periodic solution subdifferential. boundary value problem
The spectrum and an index formula for the Neumann $p-$Laplacian and multiple solutions for problems with a crossing nonlinearity
Sergiu Aizicovici Nikolaos S. Papageorgiou V. Staicu
In this paper we first conduct a study of the spectrum of the negative $p$-Laplacian with Neumann boundary conditions. More precisely we investigate the first nonzero eigenvalue. We produce alternative variational characterizations, we examine its dependence on $p\in( 1,\infty) $ and on the weight function $m\in L^{\infty}(Z) _{+}$ and we prove that the isolation of the principal eigenvalue $\lambda_{0}=0,$ is uniform for all $p$ in a bounded closed interval. All these results are then used to prove an index formula (jumping theorem) for the $d_{( S)_{+}}-$degree map at the first nonzero eigenvalue. Finally the index formula is used to prove a multiplicity result for problems with a multivalued crossing nonlinearity.
keywords: index formula homotopy invariant crossing nonlinearity. $(S) _{+}-$operator second eigenvalue
Time dependent Volterra integral inclusions in Banach spaces
Sergiu Aizicovici Yimin Ding N. S. Papageorgiou
A nonlinear Volterra inclusion associated to a family of time-dependent $m$-accretive operators, perturbed by a multifunction, is considered in a Banach space. Existence results are established for both nonconvex and convex valued perturbations. The class of extremal solutions is also investigated.
keywords: Volterra integral inclusion compact evolution operator $m$-accretive operator mild (integral) solution.
Nonlinear Dirichlet problems with double resonance
Sergiu Aizicovici Nikolaos S. Papageorgiou Vasile Staicu

We study a nonlinear Dirichlet problem driven by the sum of a $p-$Laplacian ($p>2$) and a Laplacian and which at $\pm\infty$ is resonant with respect to the spectrum of $\left(-\triangle_{p}, W_{0}^{1, p}\left(\Omega\right)\right) $ and at zero is resonant with respect to the spectrum of $\left(-\triangle, H_{0}^{1}\left(\Omega\right) \right) $ (double resonance). We prove two multiplicity theorems providing three and four nontrivial solutions respectivelly, all with sign information. Our approach uses critical point theory together with truncation and comparison techniques and Morse theory.

keywords: p-Laplacian double resonance nonlinear regularity critical groups constant sign and nodal solutions

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