Results 21 to 30 of about 3,278,971 (302)
A Counterexample in Unique Continuation [PDF]
Mathematical Research Letters, 2000 The authors prove the theorem: ``There are measurable functions \(u\), \(V\) defined on \(\mathbb{R}^2\), both supported in \(\overline B_1\), where \(B_1\) is the open unit disc, which are smooth in \(B_1\), such that \(u\), \(V\), \(Vu\in L^1(\mathbb{R}^2)\), and such that \(\Delta u-Vu= 0\) in \({\mathcal D}'\),'' thus answering a question of ...
Kenig, Carlos E., Nadirashvili, Nikolai
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Boundary Unique Continuation on $$C^1$$-Dini Domains and the Size of the Singular Set [PDF]
Archive for Rational Mechanics and Analysis, 2021 Let $u$ be a harmonic function in a $C^1$-Dini domain $D$ such that $u$ vanishes on a boundary surface ball $\partial D \cap B_{5R}(0)$. We consider an effective version of its singular set (up to boundary) $\mathcal{S}(u):=\{X\in \overline{D}: u(X ...
C. Kenig, Zihui Zhao
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Selecciones Matemáticas, 2023 We revisit the celebrated Kohn-Vogelius penalty method and discuss how to use it for the unique continuation problem where data is given in the bulk of the domain.
Erik Burman
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Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems [PDF]
Inverse Problems and Imaging, 2020 We prove a unique continuation property for the fractional Laplacian $(-\Delta)^s$ when $s \in (-n/2,\infty)\setminus \mathbb{Z}$. In addition, we study Poincar\'e-type inequalities for the operator $(-\Delta)^s$ when $s\geq 0$.
Giovanni Covi, Keijo Monkkonen, J. Railo
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Weak damping for the Korteweg-de Vries equation
Electronic Journal of Qualitative Theory of Differential Equations, 2021 For more than 20 years, the Korteweg–de Vries equation has been intensively explored from the mathematical point of view. Regarding control theory, when adding an internal force term in this equation, it is well known that the Korteweg–de Vries equation ...
Roberto de A. Capistrano Filho
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Solving ill-posed Helmholtz problems with physics-informed neural networks
Journal of Numerical Analysis and Approximation Theory, 2023 We consider the unique continuation (data assimilation) problem for the Helmholtz equation and study its numerical approximation based on physics-informed neural networks (PINNs).
Mihai Nechita
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Higher-order fractional Laplacians: An overview
Bruno Pini Mathematical Analysis Seminar, 2022 We summarize some of the most recent results regarding the theory of higher-order fractional Laplacians, i.e., the operators obtained by considering (non-integer) powers greater than 1 of the Laplace operator.
Nicola Abatangelo
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On the unique continuation of solutions to non-local non-linear dispersive equations [PDF]
Communications in Partial Differential Equations, 2020 We prove unique continuation properties of solutions to a large class of nonlinear, non-local dispersive equations. The goal is to show that if are two suitable solutions of the equation defined in such that for some non-empty open set for then for any ...
C. Kenig +3 more
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A Note on Unique Continuation for Schrodinger's Operator [PDF]
Proceedings of the American Mathematical Society, 1988 In this paper we shall prove a unique continuation theorem for Schrödinger’s operator, i ∂
Kenig, Carlos E., Sogge, Christopher D.
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Large‐Scale Analyticity and Unique Continuation for Periodic Elliptic Equations [PDF]
Communications on Pure and Applied Mathematics, 2020 We prove that a solution of an elliptic operator with periodic coefficients behaves on large scales like an analytic function in the sense of approximation by polynomials with periodic corrections.
S. Armstrong +2 more
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