On the Bernoulli free boundary problems for the half Laplacian and for the spectral half Laplacian

In this study, we investigate the application of variational methods to establish the existence of weak solutions for boundary value problems (BVPs) associated with nonlinear elliptic partial differential equations (PDEs). By employing the direct method in the calculus of variations, we reformulate the given PDEs as minimisation problems over suitable Sobolev spaces, where weak solutions correspond to critical points of appropriately defined energy functionals. The analytical framework incorporates foundational tools such as Sobolev embeddings, compactness theorems, and properties of coercivity and weak lower semi-continuity to ensure the well-posedness of the variational formulation. We present and prove an existence theorem under well-defined conditions on the nonlinear coefficients of the equations, specifically those involving the Carathéodory property, polynomial growth bounds, monotonicity, and symmetry. These conditions are shown to satisfy the hypotheses of the Fucik-Kufner theorem, which guarantees the existence - and, under convexity assumptions, uniqueness - of weak solutions. To demonstrate the utility of the theoretical results, we apply the theorem to selectednonlinear elliptic problems and verify the fulfilment of all required criteria for the existence of solutions. Through this approach, we provide a rigorous and general framework for analysing nonlinear elliptic BVPs without resorting to classical linearization techniques. The findings confirm that variational methods offer powerful, flexible tools for addressing existence questions in complex PDE systems, with relevance to mathematical modelling in physics, engineering, and applied sciences.

The initial value problem for the two-dimensional dissipative quasi-geostrophic equation derived from geophysical fluid dynamics is studied. The dissipation of this equation is given by the fractional Laplacian. It is known that the half Laplacian is a critical dissipation for the quasi-geostrophic equation. The global existence of solutions upon the suitable condition is also well known, and that solutions of a fractional dissipative equation decay with a polynomial order as the spatial variable tends to infinity. In this paper, far field asymptotics of solutions to the quasi-geostrophic equation are given in the critical and the supercritical cases. Those estimates are derived from the energy methods for the difference between the solution and its asymptotic profile.

Chapter 3 Qualitative properties of solutions to elliptic problems

In this paper we introduce a two phase version of the well-known Quadrature Domain theory, which is a generalized (sub)mean value property for (sub)harmonic functions. In concrete terms, and after reformulation into its PDE version the problem boils down to finding solution to $$ - \Delta u = (\mu_+ - \lambda_+ )\chi_{\{u > 0\}} - (\mu_- - \lambda_- )\chi_{\{u < 0\}} ~~~{\rm in }~~~ {I\!\!R}^N. $$ where \(\lambda_\pm >0 \) are given constants and \(\mu^\pm\), are non-negative bounded Radon measures, with compact support. Our primary concern is to discuss existence and geometric properties of solutions.

The authors study the existence of periodic solutions with prescribed minimal period for superquadratic and asymptotically linear autonomous second order Hamiltonian systems without any convexity assumption. Using the variational methods, an estimate on the minimal period of the corresponding nonconstant periodic solution of the above-mentioned system is obtained.

By using the tools of derivatives with respect to the domain we study the positiveness of the quadratic form a particular case of the Alt Caarelli’s functional J which related to a Bernoulli’s free boundary problem for Laplacian operator. We give sucient conditions to get local strict minimum or the stability for a Bernoulli’s free boundary problem.

The problem considered is that of minimizing the drag of a symmetric plate in infinite cavity flow under the constraints of fixed arclength and fixed chord. The flow is assumed to be steady, irrotational, and incompressible. The effects of gravity and viscosity are ignored. Using complex variables, expressions for the drag, arclength, and chord, are derived in terms of two hodograph variables, Γ (the logarithm of the speed) and β (the flow angle), and two real parameters, a magnification factor and a parameter which determines how much of the plate is a free-streamline. Two methods are employed for optimization: (1) The parameter method. Γ and β are expanded in finite orthogonal series of N terms. Optimization is performed with respect to the N coefficients in these series and the magnification and free-streamline parameters. This method is carried out for the case N = 1 and minimum drag profiles and drag coefficients are found for all values of the ratio of arclength to chord. (2) The variational method. A variational calculus method for minimizing integral functionals of a function and its finite Hilbert transform is introduced, This method is applied to functionals of quadratic form and a necessary condition for the existence of a minimum solution is derived. The variational method is applied to the minimum drag problem and a nonlinear integral equation is derived but not solved.

and H:R x R 2 ~ R is smooth. By a homoclinic or a doubly asymptotic orbit to y, where y is a periodic solution of (1), we mean a solution x + y of (1) such that I x ( 0 y(t)l--, 0 as I tl--* oo. Finding homoclinic orbits in systems like (1) can be quite difficult. In the case when y is a hyperbolic orbit this is equivalent to showing that the stable and unstable manifold of y intersect. In some situation this can be done by Melnikov's method. Recently some progress has been made by applying variational methods. Coti Zelati and Ekeland showed in [3], using dual variational methods, that (1) has at least one homoclinic orbit, if H satisfies some convexity and growth assumptions. In their proof the convexity assumption enters in a crucial way already in the set up of the variational problem. Variational methods have been used by Benci and Giannoni [2], who studied homoclinic orbits on compact Riemannian manifolds and by P. Rabinowitz in [9] who proved existence of heteroclinic orbits on the n-dimensional torus as well as their multiplicity. In [2] and [9] only special Hamiltonian of "Lagrangian" type are studied. The associated variational problem is then at least semi-definite (the

On the existence of positive solutions and solutions with compact support for a spectral nonlinear elliptic problem with strong absorption

In this paper we revisit the nonlinear Maxwell system and Maxwell-Stokes system. One of the main feature of these systems is that existence of solutions depends not only on the natural of nonlinearity of the equations, but also on the type of the boundary conditions and the topology of the domain. We review and improve our recent results on existence of solutions by using the variational methods together with modified De Rham lemmas, and the operator methods. Regularity results by the reduction method are also discussed and improved.

In this paper, we are concerned with the existence of least energy solutions of nonlinear Schrodinger equations involving the half Laplacian \begin{eqnarray} (-\Delta)^{1/2}u(x)+\lambda V(x)u(x)=u(x)^{p-1}, u(x)\geq 0, \quad x\in R^N, \end{eqnarray} for sufficiently large $\lambda$, $2 < p < \frac{2N}{N-1}$ for $N \geq 2$. $V(x)$ is a real continuous function on $R^N$. Using variational methods we prove the existence of least energy solution $u(x)$ which localize near the potential well int$(V^{-1}(0))$ for $\lambda$ large. Moreover, if the zero sets int$(V^{-1}(0))$ of $V(x)$ include more than one isolated components, then $u_\lambda(x)$ will be trapped around all the isolated components. However, in Laplacian case, when the parameter $\lambda$ large, the corresponding least energy solution will be trapped around only one isolated component and become arbitrary small in other components of int$(V^{-1}(0))$. This is the essential difference with the Laplacian problems since the operator $(-\Delta)^{1/2}$ is nonlocal.

Generic existence of Lipschitzian solutions of optimal control problems without convexity assumptions

Bernoulli's free boundary problem is an overdetermined problem in which one seeks an annular domain such that the capacitary potential satisfies an extra boundary condition. There exist two different types of solutions called elliptic and hyperbolic solutions. Elliptic solutions are ``stable'' solutions and tractable by variational methods and maximum principles, while hyperbolic solutions are ``unstable'' solutions of which the qualitative behavior is less known. We introduce a new implicit function theorem based on the parabolic maximal regularity, which is applicable to problems with loss of derivatives. Clarifying the spectral structure of the corresponding linearized operator by harmonic analysis, we prove the existence of foliated hyperbolic solutions as well as elliptic solutions in the same regularity class.

In this paper, some results concerning the existence of optimal solutions to an optimal control problem are derived. The problem involves a quasilinear hyperbolic differential equation with boundary condition and a nonlinear integral functional of action. The assumption of convexity, under which the main theorem is proved, is not connected directly with the convexity of the functional of action. In the proof, the implicit function theorem for multimappings is used.

Introduction. The problem of critical loads of a compressed orthotropic rectangular plate on an elastic base was considered. The following orthotropy parameters were set for the plate: Poisson coefficients, Young's modules for the main directions, and the shear modulus of the plate material. The components of the compressive load were uniformly distributed along two opposite edges of the plate and acted parallel to the coordinate axes. The edges of the plate were loosely pinched or pivotally supported. Cases were also considered when two parallel edges of the plate were free from loads, and the other two were freely pinched or pivotally supported.Materials and Methods. The problem was studied on the basis of a system of nonlinear Kármán-type equilibrium equations. The critical values of the load parameter were determined from a linearized problem based on a trivial solution. At the same time, the variational method in combination with the finite difference method was used to solve the boundary eigenvalue problem.Results. The problem was reduced to solving a parametric linear boundary eigenvalue problem. In case of boundary conditions of a movable hinge support, exact formulas of eigenvalues and eigenfunctions were given. While in case of free edge pinching, a variational method was used in combination with a finite-difference method, and a computer program for solving the problem was built. It was established that one or two eigenfunctions expressing the deflection of the plate could correspond to the critical value of the compressive load parameter at which the stability of the compressed plate was lost. The results of numerical calculations of the critical values of the compressive load at different values of the orthotropy parameters were presented, and graphs of the corresponding equilibrium forms were constructed. For the case of a long orthotropic plate on an elastic base, it was established that the main term of the asymptotic expansion of the solution to the linear eigenvalue problem was determined from the problem of critical loads of a compressed beam on an elastic base with an elastic modulus that coincides with the elastic modulus of the plate in the longitudinal direction.Discussion and Conclusions. The problem of critical loads of an orthotropic plate compressed in two directions lying on an elastic base was investigated. As the compressive load component increased along one direction, the critical value of the load compressing the plate along the other direction decreased. If an orthotropic plate was compressed by a load along a direction that corresponded to a greater bending stiffness, then the critical value of the loss of stability was greater than the critical value of the compressive load acting along the direction of a lesser bending stiffness. The presence of an elastic foundation increased the bearing capacity of the compressed plate.