Narrow Results

This paper proves the existence and regularity of weak solutions for a class of mixed local–nonlocal problems with singular nonlinearities. We examine both the purely singular problem and perturbed singular problems. A central contribution of this work is the inclusion of a variable singular exponent in the context of measure-valued data. Another notable feature is that the source terms in both the purely singular and perturbed components can simultaneously take the form of measures. To the best of our knowledge, this phenomenon is new, even in the case of a constant singular exponent.

This paper is concerned with an optimization problem related to the pseudo p-Laplacian eigenproblem, with Robin boundary conditions. The principal eigenvalue is minimized over a rearrangement class generated by a fixed positive function. Existence and optimality condition are proved. The popular case where the generator is a characteristic function is also considered. In this case the method of domain derivative is used to capture qualitative features of the optimal solutions.

The purpose of this paper is to prove some existence and non-existence theorems for the nonlinear elliptic problems of the form -Δ_pu = λk(x)u^q ± h(x)u^σ if x ∈ Ω, subject to the Dirichlet condition u = 0 on ∂Ω. In the proof of our results we use the sub-solution, super-solution methods and variational arguments.

Motivated by Gage [On an area-preserving evolution equation for plane curves, in Nonlinear Problems in Geometry, ed. D. M. DeTurck, Contemporary Mathematics, Vol. 51 (American Mathematical Society, Providence, RI, 1986), pp. 51–62] and Ma–Cheng [A non-local area preserving curve flow, preprint (2009), arXiv:0907.1430v2, [math.DG]], in this paper, an area-preserving flow for convex plane curves is presented. This flow will decrease the perimeter of the evolving curve and make the curve more and more circular during the evolution process. And finally, as t goes to infinity, the limiting curve will be a finite circle in the C^∞ metric.

The main objective in this paper is to obtain the existence results for bounded and unbounded solutions of some quasilinear elliptic systems. Related results as obtained here have been established recently in [C. O. Alves and A. R. F. de Holanda, Existence of blow-up solutions for a class of elliptic systems, Differ. Integral Eqs.26(1/2) (2013) 105–118]. Also, we present some references to give the connection between these types of problems with probability and stochastic processes, hoping that these are interesting for the audience of analysts likely to read this paper.

The aim of this paper is to present a convex curve evolution problem which is determined by both local (curvature $\kappa$) and global (area $A$) geometric quantities of the evolving curve. This flow will decrease the perimeter and the area of the evolving curve and make the curve more and more circular during the evolution process. And finally, as $t$ goes to infinity, the limiting curve will be a finite circle in the $C^{\infty }$ metric.

In this paper, we introduce two $1/{\kappa }^n$-type ($n\ge 1$) curvature flows for closed convex planar curves. Along the flows the length of the curve is decreasing while the enclosed area is increasing. Finally, the evolving curves converge smoothly to a finite circle if they do not develop singularity during the evolution process.

An optimal harvesting problem for a parabolic partial differential system modeling two subpopulations of the same species is investigated. The two subpopulations are competing for resources. Under conditions on the smallness of the time interval and certain biological parameters, existence and uniqueness of an optimal control pair are established.

We consider a model for bioremediation of a pollutant by bacteria in a well-stirred bioreactor. A key feature is the inclusion of dormancy for bacteria, which occurs when the critical nutrient level falls below a critical threshold. This feature gives a discrete component to the system due to the change in dynamics (governed by a system of ordinary differential equations between transitions) at switches to/from dormancy. After setting the problem in an appropriate state space, the control is the rate of injection of the critical nutrient and the functional to be minimized is the pollutant level at the final time and the amount of nutrient added. The existence of an optimal control and a discussion of the transitions between dormant and active states are given.

The design problem for semiconductor devices is studied via an optimal control approach for the standard drift–diffusion model. The solvability of the minimization problem is proved. The first-order optimality system is derived and the existence of Lagrange-multipliers is established. Further, estimates on the sensitivities are given. Numerical results concerning a symmetric n–p-diode are presented.

In this paper a mathematical model, consisting of nonlinear first-order ordinary and partial differential equations with initial and boundary conditions, for the dynamical behavior of multisection DFB (distributed feedback) semiconductor lasers is investigated. We introduce a suitable weak formulation and prove existence, uniqueness and regularity properties of the solutions. The assumptions on the data are quite general, in particular, the physically relevant case of piecewise smooth, but discontinuous with respect to space and time coefficients in the equations and in the boundary conditions is included.

In this paper we consider the Rational Large Eddy Simulation model recently introduced by Galdi and Layton. We briefly present this model, which (in principle) is similar to others commonly used, and we prove the existence and uniqueness of a class of strong solutions. Contrary to the gradient model, the main feature of this model is that it allows a better control of the kinetic energy. Consequently, to prove existence of strong solutions, we do not need subgrid-scale regularization operators, as proposed by Smagorinsky. We also introduce some breakdown criteria that are related to the Euler and Navier–Stokes equations.

In this paper, we present a tsunami model based on the displacement of the lithosphere and the mathematical and numerical analysis of this model. More precisely, we give an existence and uniqueness result for a problem which models the flow and formation of waves at the time of a submarine earthquake in the vicinity of the coast. We propose a model which describes the behavior of the fluid using a bi-dimensional shallow-water model by means of a depth-mean velocity formulation. The ocean is coupled to the Earth's crust whose movement is assumed to be controlled on a large scale by plate equations. Finally, we give some numerical results showing the formation of a tsunami.

We give the formulation of the boundary layer problem of triple deck type with a known displacement in von Mises variables. The condition associated with the displacement is transformed into a nonlocal condition. We introduce an appropriate method to prove the existence of a solution. It relies on a semi-discrete problem in which the pressure gradients are considered as a parameter. We prove the existence of a solution of the von Mises problem for Lipschitzian nondecreasing displacements. We can apply the inverse von Mises transform using an original expression of y and we prove that the functions u, v, p satisfy the system in physical variables except v(x, 0) = 0 because of a lack of regularity. We obtain all the asymptotic behaviors when y → +∞.

In this paper, we address the problem of existence, approximation, and uniqueness of solutions to an abstract doubly nonlinear equation, modeling a rate-independent process with hysteretic behavior. Models of this kind arise in, e.g., plasticity, solid phase transformations, and several other problems in non smooth mechanics. Existence of solutions is proved via passage to the limit in a time-discretization scheme, whereas uniqueness results are obtained by means of convex analysis techniques.

When solving numerically approximations of the Vlasov–Maxwell equations, the source terms in Maxwell's equations coming from the numerical solution of the Vlasov equation do not generally satisfy the continuity equation which is required for Maxwell's equations to be well-posed. Hence it is necessary to introduce generalized Maxwell's equations which remain well-posed when there are errors in the sources. Different such formulations have been introduced previously. The aim of this paper is to perform their mathematical analysis and verify the existence and uniqueness of the solution.

We consider the following nonlinear parabolic system

This paper addresses a three-dimensional model for isothermal stress-induced transformation in shape-memory polycrystalline materials. We treat the problem within the framework of the energetic formulation of rate-independent processes and investigate existence and continuous dependence issues at both the constitutive relation and quasi-static evolution level. Moreover, we focus on time and space approximation as well as on regularization and parameter asymptotics.

We prove the global existence and uniqueness of strong solutions for a compressible multifluid described by the barotropic Navier–Stokes equations in dim = 1. The result holds when the diffusion coefficient depends on the pressure. It relies on a global control in time of the L² norm of the space derivative of the density, via a new kind of entropy.

We establish new lower semicontinuity results for energy functionals containing a very general volume term of polyconvex type and a surface term depending on the spatial variable in a discontinuous way.