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We derive a geometric necessary and sufficient condition for the existence of solutions to a global eikonal equation. We also study the existence of a minimal solution to this equation, and its relation with the well-known minimal time function. 相似文献
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We study structural stability of smoothness of the maximal solution to the geometric eikonal equation on (Rd,G), d?2. This is within the framework of order zero metrics G. For a subclass of these metrics we show existence, stability as well as precise asymptotics for derivatives of the solution. These results are applicable to examples arising in Schrödinger operator theory. 相似文献
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By means of the Reilly formula and the Alexandrov maximum principle, we obtain the local C1,1 estimates of the W2,p strong solutions to the Hessian quotient equations for p sufficiently large, and then prove that these solutions are smooth. There are counterexamples to show that the integral exponent p is optimal in some cases. We modify partially the known result in the Hessian case, and extend the regularity result in the special Lagrangian case to the Hessian quotient case. 相似文献
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In this paper, we establish rectifiability of the jump set of an S
1–valued conservation law in two space–dimensions. This conservation law is a reformulation of the eikonal equation and is
motivated by the singular limit of a class of variational problems. The only assumption on the weak solutions is that the
entropy productions are (signed) Radon measures, an assumption which is justified by the variational origin. The methods are
a combination of Geometric Measure Theory and elementary geometric arguments used to classify blow–ups.?The merit of our approach
is that we obtain the structure as if the solutions were in BV, without using the BV–control, which is not available in these
variationally motivated problems.
Received June 24, 2002 / final version received November 12, 2002?Published online February 7, 2003 相似文献
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Exact solutions of Kolmogorov-Petrovskii-Piskunov equation using the modified simple equation method
The modified simple equation method is employed to find the exact solutions of the nonlinear Kolmogorov-Petrovskii-Piskunov (KPP) equation. When certain parameters of the equations are chosen to be special values, the solitary wave solutions are derived from the exact solutions. It is shown that the modified simple equation method provides an effective and powerful mathematical tool for solving nonlinear evolution equations in mathematical physics. 相似文献
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Thierry Cazenave Flvio Dickstein Fred B. Weissler 《Journal of Mathematical Analysis and Applications》2009,360(2):537-547
In this paper, we consider the nonlinear heat equation(NLH)
ut−Δu=|u|αu,