Partial Diﬀerential Equations

By Edgar Solomonik. 2000

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Citations in the corpus

Physics informed deep learning for computational elastodynamics without labeled data (2020, background): “This owes to the nature of hyperbolic PDEs as further explained in Remark 1, which has been elaborated in [59].”
Regularity for minimizers for functionals of double phase with variable exponents (2019, methodology): “If we follow the standard textbooks, Dacorogna [49], Evans [50], Gilberg-Trudinger [51], etc.”
Optimal Local Multi-scale Basis Functions for Linear Elliptic Equations with Rough Coefficient (2015, unknown): “…∈ H − 1 (Ω) to the solution u ( x ) ∈ H 10 (Ω), and I L 2 (Ω) H − 1 (Ω) is the embedding operator from L 2 (Ω) to H − 1 (Ω). see → From (1.3), we can that L − 1 is continuous and indeed a homo-morphism, and the compactness of I L 2 (Ω) H − 1 (Ω) is well known based on the Sobolev space theory [16].”
MULTI-DIMENSIONAL MORSE INDEX THEOREMS AND A SYMPLECTIC VIEW OF ELLIPTIC BOUNDARY VALUE PROBLEMS (2011, background): “As given in [21], the monotonicity of the eigenvalue of a “pure” Laplacian for both Dirichlet and Neumann boundary conditions has been established, 1 but for a general second-order elliptic operator A , the monotonicity of the eigenvalue with respect to the domain is not true for the Neumann…”
Finite Difference Methods for Differential Equations (2005, methodology): “See [Kev90], [KC81] for more detailed discussions of singular perturbation problems.”
On the classification of gradient Ricci solitons (2007, background): “However, this last case is impossible since if 1 is constant 0 = f 1 1: Knowing that 1 + + k 0; now allows us to apply the strong minimum principle to show that, if 1 + + k vanishes at some point, then it vanishes everywhere (see [22] page 244).”
Fractional powers of second order partial differential operators: extension problem and regularity theory (2010, background): “This requirement is ful lled, for instance, by an elliptic operator on a bounded domain [35, 40] or, more generally, by operators that give rise to orthogonal expansions [81, 55].”
The one‐dimensional chemotaxis model: global existence and asymptotic profile (2004, methodology): “STATEMENT OF THE RESULTS The whole analysis of this paper relies on the following lemma about the regularity of the solution semigroup of the heat equation, as taken from Taylor [17]: Lemma 2.”
Strict Lyapunov functions for semilinear parabolic partial differential equations (2011, unknown): “Consider e.g. [ 29, Chap.”
Posterior consistency via precision operators for Bayesian nonparametric drift estimation in SDEs (2012, methodology): “Moreover, C ∈ then the weak solution is more regular and, in fact, lies in ˙ H 2 p ( T ); see [13], Section 6.3.”
Concentration in Lotka-Volterra Parabolic or Integral Equations: A General Convergence Result (2009, unknown): “Then, after extraction of a subsequence, u ǫ converges locally uniformly to a function u ∈ C([0 , ∞ ) × R d ) , a viscosity solution to the following equation: In particular, a.e. in t , supp n ( t, · ) ⊂ { u ( t, · ) = 0 .”
Scalable Approach to Uncertainty Quantification and Robust Design of Interconnected Dynamical Systems (2011, unknown): “However, unlike the heat equation where the solution eventually converges to the ﬁrst eigenvector of the Laplacian, in the wave equation all the eigenvectors remain eternally excited [14].”
On a transport equation with nonlocal drift (2014, methodology): “One-sided conditions on the derivative of the drift velocity are known [26] to ensure the uniqueness of weak solutions to the Burgers and Hamilton-Jacobi equations (the extremal equation on the scale (1.3)).”
Constructive methods of invariant manifolds for kinetic problems (2003, methodology): “It is in the spirit of the classical Cauchy-Kovalevskaya theorem [93, 94, 95] about analytical Cauchy problem solutions with analytical data, as well as in the spirit of the classical Lyapunov auxilary theorem about analytical invariant manifolds in the neighborhood of xed point [98, 30] and H.”