Special Session 90: NONLINEAR ELLIPTIC DIFFERENTIAL EQUATIONS AND APPLICATIONS

Polynomial-type operators that preserve logarithmic functions

Laura Angeloni Department of Mathematics and Computer Science, University of Perugia Italy Co-Author(s):    Danilo Costarelli, Chiara Darielli

Abstract: In this talk we will present some approximation results by means of a new class of polynomial-type operators that generalize the classical Bernstein polynomials and preserve a logarithmic function. The starting point are the Bernstein-type exponential polynomials introduced by Aral, Cardenas-Morales and Garrancho in 2018 and the main idea is to replace the exponential weights by means of logarithmic ones. For such operators we first establish pointwise and uniform convergence as well as their relation with the classical King operators. This allows us to obtain a quantitative estimate of the approximation error and a Voronovskaja-type asymptotic result. By means of this formula, we obtain saturation results and inverse theorems. In particular, by the Voronovskaja formula, a second order differential operator naturally arises: the set of the solutions of the corresponding homogeneous ODE represents the saturation class of the considered operators. Some shape preserving properties are also discussed.

Asymptotic analysis of elastic elliptic membrane shells in adhesive contact

\\`Angel D Ar\\`os Rodr\\`{\\i}guez CITMAga, Universidade da Coru\\~na Spain Co-Author(s):

Abstract: By using asymptotic analysis arguments, we derive and justify a two-dimensional contact model for linearly elastic elliptic membrane shells in adhesive contact with a deformable foundation. The process is assumed to be quasistatic, and therefore the effects of inertia are neglected. Contact is modeled with normal compliance and the adhesion is modeled by introducing a surface auxiliary variable, the bonding function, the evolution of which is described by a nonlinear first order differential equation. To do this, we consider the three-dimensional contact problem, introduce a change of variable to curvilinear coordinates, together with the right scaling of the unknowns and parameters of the problem, and when the thickness of the shell (small parameter of the problem) tends to zero we obtain a two-dimensional limit model, which we justify with a rigorous convergence theorem. We will also show a fully discrete numerical scheme and results of some numerical simulations to test the performance of our model.

On the sub and supersolution method for nonlinear elliptic equations with a convective term.

Giuseppina G. Barletta University of Reggio Calabria Italy Co-Author(s):    E. Tornatore

Abstract: We present some existence (with also sign information) and regularity results for differential equations, in which the method of sub and supersolutions plays an important role. We focus on Dirichlet problem, for problems driven by a general differential operator, depending on $(x,u,\nabla u)$ , and with a convective term $f$. Our framework is that of Orlicz-Sobolev spaces. We present also several examples.

On the Well-posedness of Problems with Fixed Point Structure

Pasquale Candito University of Palermo Italy Co-Author(s):    Bruno Vassallo Mircea Sofonea

Abstract: Let $X$ and $Y$ be two arbitrary sets, $P\subset X \times Y $ and $S: X \rightarrow Y$ . We consider the problem of finding an element $u \in X$ such that $(u, Su) \in P$. We prove that the existence of the solution to this problem is obtained, provided that an associated operator $\Lambda$ has a fixed point. Moreover, under an additional assumption, the solution is unique if and only if the fixed point is unique. Then, in the framework of metric spaces $X$ and $Y$, we provide necessary and sufficient conditions for the convergence of an arbitrary sequence $\{u_n\} \subset X$ to the solution $u$. We also show some applications of our results in the study of nonlinear boundary value problems for partial differential equations.

On a Hinged Plate Equation: Paradoxes, Existence and Uniqueness

Cristian P Danet University of Craiova Romania Co-Author(s):    Cristian-Paul Danet

Abstract: Although fourth-order problems have granted attention even from the first decade of the 20th century, most of the literature deals with the Navier case or with Dirichlet boundary conditions,where Green`s function arguments are available. This talk is a contribution to the study of a hinged plate problem, i.e., we work under the more complicated boundary conditions called Steklov conditions. Such problems have been studied only in the last decade. Here, we mainly focus on some paradoxes (Babuska and Sapodzyan), existence via variational methods in the case of a semilinear biharmonic Steklov problem and on an uniqueness result based on a maximum principle.

Nonhomogeneous degenerate quasilinear problems with convection and intrinsic operator

Tornatore Elisabetta University of Palermo Italy Co-Author(s):

Abstract: A sub-supersolution method is used in the case of a quasilinear Dirichlet problem which exhibits convection, with an intrinsic operator, and whose principal part contains an unbounded coefficient $G(u)$ depending on the solution $u$. In particular a truncation technique leading to a priori estimates is developed, not only for the reaction term in the equation, but also for the unbounded coefficient. A different truncation method is used to study a Dirichlet problem whose equation is driven by a degenerate $p$-Laplacian with a weight depending on $x$ and on the solution and whose reaction term is a convection term. The existence of solutions is obtained together with uniform boundedness of the solution set. Joint work with professors D. Motreanu and R. Livrea

Nonlocal Carrier`s Double Phase Problem

Giuseppe Failla University of Palermo Italy Co-Author(s):    Leszek Gasi\`nski

Abstract: We investigate the existence and multiplicity of positive bounded solutions for a class of nonlocal, non-variational elliptic problems driven by a non-homogeneous operator with unbalanced growth, namely the double phase operator. The problem is characterized by the presence of a sign-changing weight function on the left-hand side. Our approach combines several techniques, including the sub- and super-solutions method, variational and truncation techniques, as well as tools from set-valued analysis. Furthermore, we analyze an associated one-dimensional fixed-point problem that allows us to prove the existence of $K>0$ pairs of positive solutions.

On the Well-posedness of Minimization Problems

Mircea Sofonea University of Perpignan France Co-Author(s):    Mircea Sofonea and Bruno Vassallo

Abstract: We consider a minimization problem $\cal P$ in a reflexive Banach space for which we introduce the concept of $\cal T$-well posedness. It is based on a new mathematical object, the so-called Tykhonov triple, and it extends the classical Tykhonov and Levitin-Polyak well-posedness concepts. We provide several examples and counter-examples, then we state and prove various well-posedness results. The proofs are based on convexity, compactness and lower semicontinuous arguments. We also discuss the problem of describing an optimal $\cal T$-well-posedness concept. Next, we consider the case of differentiable functionals defined on a real Hilbert space. Under additional assumptions, we prove a convergence criteria, that is, we state necessary and sufficient condition that guarantee the convergence of any arbitrary sequence to the solution of problem $\cal P$, assumed to be unique. We apply these abstract results to prove the convergence of the solution of a penalty problem to the solution of the original minimization problem, as the penalty parameter converges to zero.

Well-posedness Analysis of a Unilateral Contact Problem

Bruno Vassallo University of Messina Italy Co-Author(s):    Roberto Livrea and Mircea Sofonea

Abstract: We consider a mathematical model that describes the equilibrium of a nonlinear elastic membrane which can arrive in contact with a rigid obstacle, the so-called foundation. The membrane is fixed on its boundary, is acted upon by a vertical force, the contact is frictionless and is modelled with the well-known Signorini boundary condition. The novelty is that we model the material`s behaviour with a Hencky-type elastic constitutive law and, therefore, the problem is governed by a $p$-Laplacian operator. Using such a constitutive assumption makes the model nonstandard from a mechanical point of view and challenging from a mathematical point of view. We use a Weierstrass-type minimization argument to prove the unique weak solvability of the model. Then, we turn to the well-posedness analysis of the problem, which represents the main novelty of this manuscript. Thus, we introduce three well-posedness concepts, compare them and state and prove strong and weak well-posedness results. We also discuss the problem of finding an optimal well-posedness result. Finally, we apply these results to prove that the solution depends continuously with respect to the density of applied forces and the initial gap. We end our paper with an Appendix in which we present some preliminary results used in this manuscript.

Quasi-linear fractional Wentzell problems

Alejandro Velez-Santiago University of Puerto Rico - Rio Piedras Campus USA Co-Author(s):    Efren Mesino-Espinosa

Abstract: We investigate a generalized quasi-linear elliptic boundary value problem governed by the regional fractional $p$-Laplacian $(-\Delta)^s_{_{p,\Omega}}$ in $\Omega$, and generalized fractional Wentzell boundary conditions of type $$C`_{p,s}\mathcal{N}^{p`(1-s)}u+\beta|u|^{q-2} u+\Theta^{\eta}_qu\,=\,g \textrm{ on }\,\,\Gamma,$$ where $\Theta^{\eta}_q$ stands as a nonlocal fractional-type $q$-operator on $\Gamma$ (also refered as a Besov $q$-map), $C`_{p,s}\mathcal{N}^{p`(1-s)}$ denotes the fractional $p$-normal derivative operator in $\Gamma$, and $p,\,q$ are two exponents on the interior and boundary, respectively (which are in general unrelated between each other). We first show that this model equation admits a unique weak solution, which is globally bounded in $\overline{\Omega}$. Furthermore, a priori $L^{\infty}$-estimates for the difference of weak solutions are provided, with upper bound depending in the differences of the respective interior and boundary data functions. Additionally, a Weak Comparison Principle is derived, and we conclude by establishing a sort of nonlinear Fredholm Alternative related to this generalized elliptic fractional model equation.