The Cauchy problem for nonlinear elliptic equations

This paper is devoted to investigation of the Cauchy problem for nonlinear equations with a small parameter. They are actually small perturbations of linear elliptic equations in which case the Cauchy problem is ill-posed. To study the Cauchy problem we invoke purely nonlinear methods, such as successive iterations and L^q$L^q$ Sobolev spaces with large q$q$. We also discuss linearisable problems.     

An alternating method for an inverse Cauchy problem

In this paper, an iterative boundary element method based on our relaxed algorithm introduced in [8] is used to solve numerically a class of inverse boundary problems. A dynamical choice of the relaxation parameter is presented and a stopping criterion based on our theoretical results is used. The numerical results show that the algorithm produces a reasonably approximate solution and improves the rate of convergence of Kozlov's scheme [10]. This revised version was published online in June 2006 with corrections to the Cover Date.     

An inverse problem of identifying the coefficient in a nonlinear parabolic equation

This paper deals with the determination of a pair (p,u) in the nonlinear parabolic equation

u_t−u_xx+p(x)f(u)=0,     

Simultaneous identification of convection velocity and source strength in a convection–diffusion equation

In this work we are concerned with the analysis on a simultaneous finite element reconstruction of the convection velocity and source strength in a time-dependent convection–diffusion equation. The ill-posed problem is formulated into an output least-squares nonlinear minimization by an appropriately selected Tikhonov regularization. The regularizing effect and mathematical properties of the regularized system are justified and demonstrated. The nonlinear optimization problem is approximated by a fully discrete finite element method, whose convergence is rigorously established.     

CAUCHY PROBLEM FOR A NONLINEAR SINGULAR INTEGRAL-DIFFERENTIAL EVOLUTION SYSTEM

We study the Cauchy problem for a nonlinear evolution system with singularintegral differential terms, By means of some a priori estimates of the solution and the Leray-Schander‘s fixed point theorem, we prove the existence and the uniqueness theorems of the generalized global solution of the mentioned problem.     

On nonlinear boundary value problem corresponding to N-dimensional inverse spectral problem

We establish a relationship between an inverse optimization spectral problem for the N-dimensional Schrödinger equation $?\mathrm{\Delta }?+q(x)?=\lambda ?$ and a solution of the nonlinear boundary value problem $?\mathrm{\Delta }u+q(x)u=\lambda u?u^{\gamma ?1},u>0,u{|}_{?\mathrm{\Omega }}=0$. Using this relationship, we find an exact solution for the inverse optimization spectral problem, investigate its stability and obtain new results on the existence and uniqueness of the solution for the nonlinear boundary value problem.     

ON THE CAUCHY PROBLEM OF NONLINEAR DEGENERATE PARABOLIC EQUATION

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Cauchy problem for a generalized nonlinear liouville equation

We consider the Cauchy problem for a generalized Liouville equation. We study the existence, uniqueness, and absence of a global solution of this problem. We also discuss the local solvability of the problem.     

Critical eigenvalues for a nonlinear problem

We study the application, , where is the supremum of positive s such that the problem admits a solution. Where B ₁ is the unit ball in We show that is a decreasing function, with where is the unique solution of the problem . We also give the explicit solutions of the problem , when and show that . We show that the problem doesnt admit a solution. In the end, we give a numerical approximation of , when .     

Parametric nonlinear resonant Robin problems

We consider a nonlinear Robin problem driven by the p‐Laplacian. In the reaction we have the competing effects of two nonlinearities. One term is parametric, strictly $(p?1)$‐sublinear and the other one is $(p?1)$‐linear and resonant at any nonprincipal variational eigenvalue. Using variational tools from the critical theory (critical groups), we show that for all big values of the parameter λ the problem has at least five nontrivial smooth solutions.     

A convex-concave problem with a nonlinear boundary condition

In this paper we study the existence of nontrivial solutions of the problem

     

A two-stage method for nonlinear inverse problems

In this work we are interested in the solution of nonlinear inverse problems of the form F(x)=y$F(x)=y$. We consider a two-stage method which is third order convergent for well-posed problems. Combining the method with Levenberg–Marquardt regularization of the linearized problems at each stage and using the discrepancy principle as a stopping criterion, we obtain a regularization method for ill-posed problems. Numerical experiments on some parameter identification and inverse acoustic scattering problems are presented to illustrate the performance of the method.     

Stability analysis in the inverse Robin transmission problem

In this paper, we consider the conductivity problem with piecewise‐constant conductivity and Robin‐type boundary condition on the interface of discontinuity. When the quantity of interest is the jump of the conductivity, we perform a local stability estimate for a parameterized non‐monotone family of domains. We give also a quantitative stability result of local optimal solution with respect to a perturbation of the Robin parameter. In order to find an optimal solution, we propose a Kohn–Vogelius‐type cost functional over a class of admissible domains subject to two boundary values problems. The analysis of the stability involves the computation of first‐order and second‐order shape derivative of the proposed cost functional, which is performed rigorously by means of shape‐Lagrangian formulation without using the shape sensitivity of the states variables. © 2016 The Author. Mathematical Methods in the Applied Sciences Published by John Wiley & Sons Ltd.     

Gradient computation in a nonlinear inverse problem

This paper deals with a nonlinear inverse problem to determine the Neumann condition on the boundary $\text{Γ}{}_{\mathrm{L}}\text{??Ω}$, from measurements in the domain $\text{Ω}$. This condition is characterised by the width of Γ_L and by the constant value of the flux on this boundary. The direct problem is the Laplacian problem corresponding to flow modelling in a confined aquifer and Γ_L corresponds to the contact with a fault. Some properties of associated direct application are given and in particular, we show how one can compute its gradient by some explicit formulas. To cite this article: D.-G. Calugaru, J.-M. Crolet, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

On the stability of the positive steady states for a nonhomogeneous semilinear Cauchy problem

This paper is contributed to the Cauchy problem

     

Existence of solutions for a nonlinear PDE with an inverse square potential

Via Linking theorem and delicate energy estimates, the existence of nontrivial solutions for a nonlinear PDE with an inverse square potential and critical sobolev exponent is proved. This result gives a partial (positive) answer to an open problem proposed in Ferrero and Gazzola (J. Differential Equations 177 (2001) 494).     

Coefficient estimation in a first-order nonlinear hyperbolic Cauchy problem

This paper deals with the estimation and approximation of coefficient function in a first-order, nonlinear, hyperbolic Cauchy problem. The estimation is accomplished by minimizing a functional which measures the error between a finite set of given observations and the corresponding values of the solution generated by the coefficient function. A class of admissible coefficient functions is defined, and it is proved that minimizing coefficient function always exists within this class. We also develop an approximation by a sequence of solutions of associated finite-dimensional minimization problems.