This article proves a case of the p-adic Birch and Swinnerton–Dyer conjecture for Garrett p-adic L-functions of (Bertolini et al. in On p-adic analogues of the Birch and Swinnerton–Dyer conjecture for Garrett L-functions, 2021), in the imaginary dihedral exceptional zero setting of extended analytic rank 4.
We consider motion by anisotropic curvature of a network of three curves immersed in the plane meeting at a triple junction and with the other ends fixed. We show existence, uniqueness and regularity of a maximal geometric solution and we prove that, if the maximal time is finite, then either the length of one of the curves goes to zero or the (L^2)-norm of the anisotropic curvature blows up.
In the context of the Calculus of Variations for non-convex, vector variational problems, the natural process of going from a function (phi ) to its quasiconvexification (Qphi ) is quite involved, and, most of the time, an impossible task. We propose to look at the reverse process, what might be called inverse quasiconvexification: start from a function (phi _0), and find functions (phi ) for which (phi _0=Qphi ). In addition to establishing a few general principles, we show several explicit examples motivated by their application to inverse problems in conductivity.
In this note, we prove a blow-up result for a semilinear generalized Tricomi equation with nonlinear term of derivative type, i.e., for the equation (mathcal {T}_ell u=|partial _t u|^p), where (mathcal {T_ell }=partial _t^2-t^{2ell }Delta ). Smooth solutions blow up in finite time for positive Cauchy data when the exponent p of the nonlinear term is below (frac{mathcal {Q}}{mathcal {Q} -2}), where (mathcal {Q}=(ell +1)n+1) is the quasi-homogeneous dimension of the generalized Tricomi operator (mathcal {T}_ell ). Furthermore, we get also an upper bound estimate for the lifespan.
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