255 research outputs found
Quasilinear problems involving a perturbation with quadratic growth in the gradient and a noncoercive zeroth order term
In this paper we consider the problem u in H^1_0 (Omega), - div (A(x) Du) =
H(x, u, Du) + f(x) + a_0 (x) u in D'(Omega), where Omega is an open bounded set
of R^N, N \geq 3, A(x) is a coercive matrix with coefficients in
L^\infty(Omega), H(x, s, xi) is a Carath\'eodory function which satisfies for
some gamma > 0 -c_0 A(x) xi xi \leq H(x, s, xi) sign (s) \leq gamma A(x) xi xi
a.e. x in Omega, forall s in R, forall xi in R^N, f belongs to L^{N/2} (Omega),
and a_0 \geq 0 to L^q (Omega ), q > N/2. For f and a_0 sufficiently small, we
prove the existence of at least one solution u of this problem which is
moreover such that e^{delta_0 |u|} - 1 belongs to H^1_0 (Omega) for some
delta_0 \geq gamma, and which satisfies an a priori estimate.Comment: 37 pages, 2 figure
Existence of Weak Efficient Solutions of Set-Valued Optimization Problems
In this paper, we consider a new scalarization function for set-valued maps.
As the main goal, by using this scalarization function, we obtain some
Weierstrass-type theorems for the noncontinuous set optimization problems via
the coercivity and noncoercivity conditions. This contribution improves various
existing results in the literature
Stabilised finite element methods for ill-posed problems with conditional stability
In this paper we discuss the adjoint stabilised finite element method
introduced in, E. Burman, Stabilized finite element methods for nonsymmetric,
noncoercive and ill-posed problems. Part I: elliptic equations, SIAM Journal on
Scientific Computing, and how it may be used for the computation of solutions
to problems for which the standard stability theory given by the Lax-Milgram
Lemma or the Babuska-Brezzi Theorem fails. We pay particular attention to
ill-posed problems that have some conditional stability property and prove
(conditional) error estimates in an abstract framework. As a model problem we
consider the elliptic Cauchy problem and provide a complete numerical analysis
for this case. Some numerical examples are given to illustrate the theory.Comment: Accepted in the proceedings from the EPSRC Durham Symposium Building
Bridges: Connections and Challenges in Modern Approaches to Numerical Partial
Differential Equation
Stabilised finite element methods for non-symmetric, non-coercive and ill-posed problems. Part I: elliptic equations
In this paper we propose a new method to stabilise non-symmetric indefinite
problems. The idea is to solve a forward and an adjoint problem simultaneously
using a suitable stabilised finite element method. Both stabilisation of the
element residual and jumps of certain derivatives of the discrete solution over
element faces may be used. Under the assumption of well posedness of the
partial differential equation and its associated adjoint problem we prove
optimal error estimates in and norms in an abstract framework. Some
examples of problems that are neither symmetric nor coercive, but that enter
the abstract framework are given. First we treat indefinite
convection-diffusion equations, with non-solenoidal transport velocity and
either pure Dirichlet conditions or pure Neumann conditions and then a Cauchy
problem for the Helmholtz operator. Some numerical illustrations are given.Comment: Second part in preparation: Stabilised finite element methods for
non-symmetric, non-coercive and ill-posed problems. Part II: hyperbolic
equation
SINGULAR NEUMANN PROBLEMS AND LARGE-TIME\ud BEHAVIOR OF SOLUTIONS OF NONCOERCIVE\ud HAMILTON-JACOBI EQUATIONS
We investigate the large-time behavior of viscosity solutions of Hamilton- Jacobi equations with noncoercive Hamiltonian in a multidimensional Euclidean space. Our motivation comes from a model describing growing faceted crystals recently discussed by E. Yokoyama, Y. Giga and P. Rybka. Surprisingly, growth rates of viscosity solutions of these equations depend on x-variable. In a part of the space called the effective domain, growth rates are constant but outside of this domain, they seem to be unstable. Moreover, on the boundary of the effective domain, the gradient with respect to x-variable of solutions blows up as time goes to infinity. Therefore, we are naturally led to study singular Neumann problems for stationary Hamilton-Jacobi equations. We establish the existence, stability and comparison results for singular Neumann problems and apply the results for a large-time asymptotic profile on the effective domain of viscosity solutions of Hamilton-Jacobi equations with noncoercive Hamiltonian
Robust Superlinear Krylov Convergence for Complex Noncoercive Compact-Equivalent Operator Preconditioners
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