关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

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Methods Of Modern Mathematical Physics

5. M. Green, J. Schwarz, and E. Witten, Superstring theory, Cambridge Univ. Press, 1987. 6. J. Isenberg, P. Yasskin, and P. Green, Non-self-dual gauge fields, Phys. Lett. 78B (1978), 462-464. 7. B. Kostant, Graded manifolds, graded Lie theory, and prequantization, Differential Geometric Methods in Mathematicas Physics, Lecture Notes in Math., vol. 570, SpringerVerlag, Berlin and New York, 1977. 8. C. LeBrun, Thickenings and gauge fields, Class. Quantum Grav. 3 (1986), 1039-1059. 9. , Thickenings and conformai gravity, preprint, 1989. 10. C. LeBrun and M. Rothstein, Moduli of super Riemann surfaces, Commun. Math. Phys. 117(1988), 159-176. 11. Y. Manin, Critical dimensions of string theories and the dualizing sheaf on the moduli space of (super) curves, Funct. Anal. Appl. 20 (1987), 244-245. 12. R. Penrose and W. Rindler, Spinors and space-time, V.2, spinor and twistor methods in space-time geometry, Cambridge Univ. Press, 1986. 13. R. Ward, On self-dual gauge fields, Phys. Lett. 61A (1977), 81-82. 14. E. Witten, An interpretation of classical Yang-Mills theory, Phys. Lett. 77NB (1978), 394-398. 15. , Twistor-like transform in ten dimensions, Nucl. Phys. B266 (1986), 245-264. 16. , Physics and geometry, Proc. Internat. Congr. Math., Berkeley, 1986, pp. 267302, Amer. Math. Soc, Providence, R.I., 1987.

Infinite-Dimensional Dynamical Systems in Mechanics and Physics (Roger Temam)

Reaction-diffusion waves in biology.

We extend Cordero-Erausquin et al.’s Riemannian Borell–Brascamp–Lieb inequality to Finsler manifolds. Among applications, we establish the equivalence between Sturm, Lott and Villani’s curvature-dimension condition and a certain lower Ricci curvature bound. We also prove a new volume comparison theorem for Finsler manifolds which is of independent interest.

/pdf/finsler-interpolation-inequalities-4foj60gmyj.pdf

Finsler interpolation inequalities

We study the spectrum of the linearized NLS equation in three dimensions in association with the energy spectrum. We prove that unstable eigenvalues of the linearized NLS problem are related to negative eigenvalues of the energy spectrum, while neutrally stable eigenvalues may have both positive and negative energies. The nonsingular part of the neutrally stable essential spectrum is always related to the positive energy spectrum. We derive bounds on the number of unstable eigenvalues of the linearized NLS problem and study bifurcations of embedded eigenvalues of positive and negative energies. We develop the L 2 scattering theory for the linearized NLS operators and recover results of Grillakis [5] with a Fermi golden rule. c

https://dmpeli.math.mcmaster.ca/PaperBank/spectrumCPV.pdf

Spectra of Positive and Negative Energies in the Linearized NLS Problem

Some models in population dynamics with intra-specific competition lead to integro-differential equations where the integral term corresponds to nonlocal consumption of resources [8][9]. The principal difference of such equations in comparison with traditional reaction-diffusion equation is that homogeneous in space solutions can lose their stability resulting in emergence of spatial or spatio-temporal structures [4]. We study the existence and global bifurcations of such structures. In the case of unbounded domains, transition between stationary solutions can be observed resulting in propagation of generalized travelling waves (GTW). GTWs are introduced in [18] for reaction-diffusion systems as global in time propagating solutions. In this work their existence and properties are studied for the integro-differential equation. Similar to the reaction-diffusion equation in the monostable case, we prove the existence of generalized travelling waves for all values of the speed greater or equal to the minimal one. We illustrate these results by numerical simulations in one and two space dimensions and observe a variety of structures of GTWs.

/pdf/spatial-structures-and-generalized-travelling-waves-for-an-2lfb2dh80q.pdf

Spatial structures and generalized travelling waves for an integro-differential equation

We consider a spinless particle coupled to a photon field and prove that even if the Schrodinger operator p2+V does not have eigenvalues the system can have a ground state. We describe the coupling by means of the Pauli-Fierz Hamiltonian and our result holds in the case where the coupling constant α is small.

Enhanced Binding in Non-Relativistic QED

We obtain solvability conditions in \(H^{2}\) for some elliptic equations in cylindrical domains using the methods of spectral theory and scattering theory for Schrodinger type operators. We prove the existence of standing solitary waves in \(H^{2}\) for some nonlinear equations. Both linear and nonlinear problems involve second order differential operators without Fredholm property.

Solvability conditions for some linear and nonlinear non-Fredholm elliptic problems

We show the existence of stationary solutions for some reaction-diffusion type equations in the appropriate H2 spaces using the fixed point technique when the elliptic problem contains second order differential operators with and without Fredholm property.

https://hal.archives-ouvertes.fr/hal-00653648/document

Vitali Vougalter

Papers

Spectra of Positive and Negative Energies in the Linearized NLS Problem

Spatial structures and generalized travelling waves for an integro-differential equation

Enhanced Binding in Non-Relativistic QED

Solvability conditions for some linear and nonlinear non-Fredholm elliptic problems

On the existence of stationary solutions for some non-Fredholm integro-differential equations