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We study the large time behaviour of the solution of linear dispersive partial differential equations posed on a finite interval, when at least one of the prescribed boundary conditions is time periodic. We use the Q equation approach, pioneered in Fokas & Lenells 2012 and applied to linear problems on the half-line in Fokas & van der Weele 2021, to characterise necessary conditions for the solution of such problem to be periodic, at least in an asymptotic sense. We then fully describe the periodicity properties of the solution in three important illustrative examples, recovering known results for the second-order cases and establishing new results for the third order case.
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In this paper, we consider a family of second-order elliptic systems subject to a periodically oscillating Robin boundary condition. We establish the qualitative homogenization theorem on any Lipschitz domains satisfying a non-resonance condition. We also use the quantitative estimates of oscillatory integrals to obtain the dimension-dependent convergence rates in $L^2$, assuming that the domain is smooth and strictly convex.
We prove the existence of unique solutions to the Dirichlet boundary value problems for linear second-order uniformly parabolic operators in either divergence or non-divergence form with boundary blowup low-order coefficients. The domain is possibly time varying, non-smooth, and satisfies the exterior measure condition.
Discrete approximations to the equation begin{equation*} L_{cont}u = u^{(4)} + D(x) u^{(3)} + A(x) u^{(2)} + (A(x)+H(x)) u^{(1)} + B(x) u = f, ; xin[0,1] end{equation*} are considered. This is an extension of the Sturm-Liouville case $D(x)equiv H(x)equiv 0$ [ M. Ben-Artzi, J.-P. Croisille, D. Fishelov and R. Katzir, Discrete fourth-order Sturm-Liouville problems, IMA J. Numer. Anal. {bf 38} (2018), 1485-1522. doi: 10.1093/imanum/drx038] to the non-self-adjoint setting. The natural boundary conditions in the Sturm-Liouville case are the values of the function and its derivative. The inclusion of a third-order discrete derivative entails a revision of the underlying discrete functional calculus. This revision forces evaluations of accurate discrete approximations to the boundary values of the second, third and fourth order derivatives. The resulting functional calculus provides the discrete analogs of the fundamental Sobolev properties--compactness and coercivity. It allows to obtain a general convergence theorem of the discrete approximations to the exact solution. Some representative numerical examples are presented.
Let $Omega subset {mathbb R}^N$ ($N geq 3$) be a $C^2$ bounded domain and $F subset partial Omega$ be a $C^2$ submanifold of dimension $0 leq k leq N-2$. Put $delta_F(x)=dist(x,F)$, $V=delta_F^{-2}$ in $Omega$ and $L_{gamma V}=Delta + gamma V$. Denote by $C_H(V)$ the Hardy constant relative to $V$ in $Omega$. We study positive solutions of equations (LE) $-L_{gamma V} u = 0$ and (NE) $-L_{gamma V} u+ f(u) = 0$ in $Omega$ when $gamma < C_H(V)$ and $f in C({mathbb R})$ is an odd, monotone increasing function. We establish the existence of a normalized boundary trace for positive solutions of (LE) - first studied by Marcus and Nguyen for the case $F=partial Omega$ - and employ it to investigate the behavior of subsolutions and super solutions of (LE) at the boundary. Using these results we study boundary value problems for (NE) and derive a-priori estimates. Finally we discuss subcriticality of (NE) at boundary points of $Omega$ and establish existence and stability results when the data is concentrated on the set of subcritical points.
In this paper we develop the global symbolic calculus of pseudo-differential operators generated by a boundary value problem for a given (not necessarily self-adjoint or elliptic) differential operator. For this, we also establish elements of a non-self-adjoint distribution theory and the corresponding biorthogonal Fourier analysis. We give applications of the developed analysis to obtain a-priori estimates for solutions of operators that are elliptic within the constructed calculus.
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