Pointwise Estimates for Laplace Equation. Applications to the Free Boundary of the Obstacle Problem with Dini Coefficients

In this paper we are interested in pointwise regularity of solutions to elliptic equations. In a first result, we prove that if the modulus of mean oscillation of Δu at the origin is Dini (in L ^p average), then the origin is a Lebesgue point of continuity (still in L ^p average) for the second derivatives D ² u. We extend this pointwise regularity result to the obstacle problem for the Laplace equation with Dini right hand side at the origin. Under these assumptions, we prove that the solution to the obstacle problem has a Taylor expansion up to the order 2 (in the L ^p average). Moreover we get a quantitative estimate of the error in this Taylor expansion for regular points of the free boundary. In the case where the right hand side is moreover double Dini at the origin, we also get a quantitative estimate of the error for singular points of the free boundary. Our method of proof is based on some decay estimates obtained by contradiction, using blow-up arguments and Liouville Theorems. In the case of singular points, our method uses moreover a refined monotonicity formula.      

Fujita Exponent for Porous Medium Equation with Convection and Nonlinear Boundary Condition

This paper is concerned with the critical exponent of the porous medium equation with convection and nonlinear boundary condition. It is shown that the coefficient of the lower order term is an important factor that determines the critical exponent.     

On the Higher-Order Method for the Solution of a Nonlinear Scalar Equation

In this paper, a higher-order method for the solution of a nonlinear scalar equation is presented. It is proved that the new method is locally convergent with an order of (m+2), where m is the highest order derivative used in the iterative formula. Some numerical examples are used to demonstrate the new method.     

Weighted Energy Decay for 1D Klein–Gordon Equation

We obtain a dispersive long-time decay in weighted energy norms for solutions of the 1D Klein–Gordon equation with generic potential. The decay extends the results obtained by Jensen, Kato and Murata for the equations of Schrödinger's type by the spectral approach. For the proof we modify the approach to make it applicable to relativistic equations.     

Global Existence and Exponential Stability for a Quasilinear Wave Equation with Memory Damping at the Boundary

In this paper, we focus on the global well-posedness of a quasilinear wave equation with a memory boundary condition. Under conditions on the geometry of the domain and the relaxation function describing the memory properties of the boundary, we obtain the existence, regularity and uniqueness of the global solution to the system. We prove also that the energy of the global solution to the system decays exponentially. The research was supported by the National Natural Science Foundation of China under Grant 60504001, Programa Nacional de Ayudas Para la Movilidad under Grant SB2003-0271 in Spain and the SIMUMAT projet of the CAM (Spain). The author is grateful to Prof. Enrique Zuazua for fruitful discussion.     

Global Periodic Attractor for Strongly Damped and Driven Wave Equations

In this paper we consider the strongly damped and driven nonlinear wave equations under homogeneous Dirichlet boundary conditions. By introducing a new norm which is equivalent to the usual norm, we obtain the existence of a global periodic attractor attracting any bounded set exponentially in the phase space, which implies that the system behaves exactly as a one dimensional system.     

Least-Squares Solution with the Minimum-Norm for the Matrix Equation A^TXB ＋ B^TX^TA = D and Its Applications

An efficient method based on the projection theorem,the generalized singular value decompositionand the canonical correlation decomposition is presented to find the least-squares solution with the minimum-norm for the matrix equation A~TXB B~TX~TA=D.Analytical solution to the matrix equation is also derived.Furthermore,we apply this result to determine the least-squares symmetric and sub-antisymmetric solution ofthe matrix equation C~TXC=D with minimum-norm.Finally,some numerical results are reported to supportthe theories established in this paper.     

Asymptotic Behaviour for a Nonlinear Schrödinger Equation in Domains with Moving Boundaries

We consider a nonlinear Schrödinger equation in a time-dependent domain Q _τ of ?² given by $$u_{\tau} - i u_{\varepsilon\varepsilon} + |u|^{2} u + \gamma v=0. $$ We prove the well-posedness of the above model and analyze the behaviour of the solution as t→+∞. We consider two situations: the conservative case (γ=0) and the dissipative case (γ>0). In both situations the existence of solutions are proved using the Galerkin method and the stabilization of solutions are obtained considering multiplier techniques.     

Multisymplectic Structure and Multisymplectic Scheme for the Nonlinear Ware Equation

Abstract The multisvmplectic structure of the nonlinear wave equation is derived directly from the variationalprinciple. In the numerical aspect,we present a multisymplectic nine points scheme which is equivalent to themultisymplectic Preissman scheme.A series of numerical results are reported to illustrate the effectiveness ofthe scheme.     

On Integrability of the Camassa–Holm Equation and Its Invariants

Using geometrical approach exposed in (Kersten et al. in J. Geom. Phys. 50:273–302, [2004] and Acta Appl. Math. 90:143–178, [2005]), we explore the Camassa–Holm equation (both in its initial scalar form, and in the form of 2×2-system). We describe Hamiltonian and symplectic structures, recursion operators and infinite series of symmetries and conservation laws (local and nonlocal). This work was supported in part by the NWO–RFBR grant 047.017.015 and RFBR–Consortium E.I.N.S.T.E.I.N. grant 06-01-92060.     

Dressing for a Novel Integrable Generalization of the Nonlinear Schrödinger Equation

We implement the dressing method for a novel integrable generalization of the nonlinear Schrödinger equation. As an application, explicit formulas for the N-soliton solutions are derived. As a by-product of the analysis, we find a simplification of the formulas for the N-solitons of the derivative nonlinear Schrödinger equation given by Huang and Chen.     

Stabilization for a Fourth Order Nonlinear Schrödinger Equation in Domains with Moving Boundaries

In the present paper we study the well-posedness using the Galerkin method and the stabilization considering multiplier techniques for a fourth-order nonlinear Schrödinger equation in domains with moving boundaries. We consider two situations for the stabilization: the conservative case and the dissipative case.     

A Mini Max Theorem for the Functionals with Hemicontinuous Gateaux Derivative and the Solution of the Boundary Value Problem for the Nonlinear Wave Equation

In this paper, a mini max theorem was showed mega which the paper proves a new existent and unique result on solution of the boundary value problem for the nonlinear wave equation by using the mini max theorem.     

An Inequality for Matrices and Its Applications in Differential Geometry

For a matrix A=(α_(ij))we denote by N(A)the square of the norm of A,i.6., N(A)=trace A~t A=∑(α_(ij))~2.In this paper we prove the following inequality. Theorem 1 Let A_1,A_2,…,A_p be symmetric(n×n)-matrices(p≥2).We denoteS_(αβ)=trace A_α~tA_β,S_α=S_(αα)=N(A_α),S=S_1+…+S_p。Then     

Estimate on the Dimension of Global Attractor for Nonlinear Dissipative Kirchhoff Equation

In this paper,we estimate the dimension of the global attractor for nonlinear dissipative Kirchhoff equation in Hilbert spaces H ₀¹×L ²(Ω) and D(A)×H ₀¹(Ω). Using rescaling technology and linear variation method, we obtain the upper bound for its Hausdorff and fractal dimensions.     

Boundary Control Problem for a Nonlinear Convection–Diffusion–Reaction Equation

Computational Mathematics and Mathematical Physics - The solvability of boundary-value and extremum problems for a nonlinear convection–diffusion–reaction equation with mixed boundary...     

On the Functional Estimation and Its Applications

In this paper, the following results are obtained. The functional estimation theorem: Let X, Y be linear spaces, normedbu ‖·‖_X, ‖·‖_Y, respectively: X be a subspace of X: Y(?)Y. Suppose that F is a functional on X×Y to [0, ∞). which has the properties : F(f_1 f_2, g) F(f_1, g) F(f_2, g), and F(f, g)相似文献   

Initial Boundary Value Problem for A Generalized Nonlinear Telegraph Equation with Nonlinear Damping北大核心CSCD

In this paper, the existence and uniqueness of the global generalized solution and the global classical solution of the initial boundary value problem for the generalized nonlinear telegraph equation with nonlinear damping © 2022 Chinese Academy of Sciences. All rights reserved.     

Hyperplane Envelopes and the Clairaut Equation

The subject of envelopes has been part of differential geometry from the beginning. This paper brings a modern perspective to the classical problem of envelopes of families of affine hyperplanes. In the process, the classical results are generalized and unified.     

Explicit Dimensional Formula of Atractor for Damped Sine-Gordon Equation with Neumann or Periodic Boundary Conditions

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