# Boundary value problems for pseudo-differential equations with singular symbols

• Sabir Umarov
Part of the Developments in Mathematics book series (DEVM, volume 41)

## Abstract

Let $$\varOmega \subset \mathbb{R}^{n}$$ be a bounded domain with a smooth boundary or $$\varOmega = \mathbb{R}^{n}.$$ This chapter discusses well-posedness problems of general boundary value problems for pseudo-differential and differential-operator equations of the form
$$\displaystyle\begin{array}{rcl} L[u]& \equiv & \frac{\partial ^{m}u} {\partial t^{m}} +\sum _{ k=0}^{m-1}A_{ k}(t)\frac{\partial ^{k}u} {\partial t^{k}} = f(t,x),\quad t \in (T_{1},T_{2}),\ x \in \varOmega,{}\end{array}$$
(4.1)
$$\displaystyle\begin{array}{rcl} B_{k}[u]& \equiv & \sum _{j=0}^{m-1}b_{ kj}\frac{\partial ^{j}u(t_{kj},x)} {\partial t^{j}} =\varphi _{k}(x),\quad x \in \varOmega,\,k = 0,\mathop{\ldots },m - 1,{}\end{array}$$
(4.2)
where f(t, x) is defined on $$(T_{1},T_{2})\times \varOmega,$$ $$-\infty <T_{1} <T_{2} \leq \infty,$$ and $$\varphi _{k}(x),\,x \in \varOmega,\,k = 0,\ldots,m - 1,$$ are given functions; A k (t) and b kj , $$k = 0,\ldots,m - 1,\,j = 0,\ldots,m - 1,$$ are operators acting on some spaces (specified below) of functions defined on Ω; and $$t_{jk} \in [T_{1},T_{2}],\,j,k = 0,\ldots,m - 1.$$ For example, when $$\varOmega = \mathbb{R}^{n},$$ the latter operators may act as ΨDOSS defined on the space of distributions $$\varPsi _{-G,p^{'}}^{'}(\mathbb{R}^{n})$$ with an appropriate $$G \subset R^{n}$$.

## References

Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. Comm. Pure and Appl. Math., 12, 623–727 (1969)
2. [Ata02]
Atakhodzhaev, M.A.: Ill-posed internal boundary value problems for the biharmonic equation. VSP, Utrecht (2002)
3. [BRV05]
Bacchelli, B., Bozzini, M., Rabut, C., Varas, M.: Decomposition and reconstruction of multidimensional signals using polyharmonic pre-wavelets. Applied and Computational Harmonic Analysis, 18, 282–299 (2005)
4. [BJS64]
Bers, L., John, F., Schechter M.: Partial Differential Equations. Interscience Publishers, New York - London - Sydney (1964)
5. [BD64]
Björken, J., Drel, S.: Relativistic Quantum Theory, V I. McGraw Hill Book Co., New York (1964)Google Scholar
6. [Bor69]
Borok, B.M.: Uniqueness classes for the solution of a boundary problem with an infinite layer for systems of linear partial differential equations with constant coefficients. Mat. Sb., 79 (121): 2(6), 293–304 (1969)Google Scholar
7. [Bor71]
Borok, V.M.: Correctly solvable boundary-value problems in an infinite layer for systems of linear partial differential equations. Math. USSR. Izv. 5, 193–210 (1971)
8. [Du33]
Duhamel, J.M.C.: Mémoire sur la méthode générale relative au mouvement de la chaleur dans les corps solides plongés dans les milieux dont la température varie avec le temps. J. Ec. Polyt. Paris 14, Cah. 22, 20 (1833)Google Scholar
9. [Ede75]
Edenhofer, E.: Integraldarstellung einer m-polyharmonischen Funktion, deren Funktionswerte und erste m − 1 Normalableitungen auf einer Hypersphäre gegeben sind Math. Nachr. 68, 105–113 (1975)
10. [FJG08]
Feng, B., Ji, D., Ge, W.: Positive solutions for a class of boundary value problem with integral boundary conditions in Banach spaces. J. Comput. Appl. Math., 222, 351–363 (2008)
11. [Går98]
Gårding, L.: Hyperbolic equations in the twentieth century. Seminaires et Congres, 3, 37–68 (1998)Google Scholar
12. [G84]
Gorbachuk, V.I., Gorbachuk M.L.: Boundary Value Problems for Operator Differential Equations. Kluwer Academic Publishers. Netherlands (1990)Google Scholar
13. [HK07]
Haußmann, W., Kounchev, O.: On polyharmonic interpolation. J. Math. Anal. Appl. 331 (2), 840–849 (2007)
14. [HK93]
Hayman, W.K., Korenblum, B.: Representation and uniqueness theorems for polyharmonic functions. J. d’Analyse Mathématique, 60, 113–133 (1993)
15. [Hor83]
Hörmander, L.: The Analysis of Linear Partial Differential Operators, I - IV. Springer-Verlag, Berlin-Heidelberg-New-York (1983)Google Scholar
16. [Lop53]
Lopatinskiĩ, Ya.B.: On a method of reducing boundary problems for a system of differential equations of elliptic type to regular integral equations. Ukrain. Mat. Zb. 5, 123–151 (1953)Google Scholar
17. [Nai67]
Naimark, M.A.: Linear differential operators, 1,2, F. Ungar (1967) (Translated from Russian)Google Scholar
18. [Nap82]
Napalkov, V.V.: Convolution equations in multidimensional spaces. Mathematical Notes of the Acad. Sci. USSR, 25 (5), 393 pp., Springer (1979)Google Scholar
19. [N12]
Napalkov, V.V., Nuyatov, A.A.: The multipoint de la Vallée-Poussin problem for a convolution operator. Sbornik: Mathematics, 203 (2), 224–233 (2012)
20. [Naz97]
Nazarova, M. Kh.: On weak well-posedness of certain boundary value problems generated by a singular Bessel’s operator. Uzb. Math. J. 3, 63–70 (1997) (in Russian)
21. [Pet96]
Petrovskii, I.G.: Selected works: Systems of partial differential equations and algebraic geometry, Part I. Gordon and Breach Publishers (1996)Google Scholar
22. [Pok68]
Pokornyi, Yu.V.: On estimates for the Green’s function for a multipoint boundary problem. Mat. Zametki, 4 (5), 533–540 (1968)
23. [Pta84]
Ptashnik, B.I.: Ill-Posed Boundary Value Problems for Partial Differential Equations. Kiev (1984) (in Russian)Google Scholar
24. [Pul99]
Pulkina, L.S.: A non-local problem with integral conditions for hyperbolic equations. Electronic Journal of Differential Equations, 45, 1–6 (1999)
25. [Ren08]
Render, H.: Real Bargmann spaces, Fischer decompositions and Sets of uniqueness for polyharmonic functions. Duke Math. J., 142 (2), 313–352 (2008)
26. [Sam83]
Samarov, K.L.: Solution of the Cauchy problem for the Schrödinger equation for a relativistic free particle. Soviet Phys. Docl., 271 (2), 334–337 (1983)
27. [Sat59]
Sato, M.: Theory of Hyperfunctions, I. Journal of the Faculty of Science, University of Tokyo. Sect. 1, Mathematics, astronomy, physics, chemistry, 8 (1), 139–193 (1959)Google Scholar
28. [Sat60]
Sato, M.: Theory of Hyperfunctions, II. Journal of the Faculty of Science, University of Tokyo. Sect. 1, Mathematics, astronomy, physics, chemistry, 8 (2), 387–437 (1960)Google Scholar
29. [SKK73]
Sato, M., Kawai, T., Kashiwara, M.: Microfunctions and pseudo-differential equations. Lecture Notes in Mathematics, 287, Springer-Verlag, Berlin-Heidelberg-New York, 265–529 (1973)Google Scholar
30. [Say06]
Saydamatov, E.M.: Well-posedness of the Cauchy problem for inhomogeneous time-fractional pseudo-differential equations, Frac. Calc. Appl. Anal., 9(1), 1–16 (2006)
31. [Say07]
Saydamatov, E.M.: Well-posedness of general nonlocal nonhomogeneous boundary value problems for pseudo-differential equations with partial derivatives. Siberian Advances in Mathematics, 17 (3), 213–226 (2007)
32. [Sch51]
Schwartz, L.: Théorie des distributions I, II. Hermann, Paris (1951)Google Scholar
33. [Sha53]
Shapiro, Z.Ya.: On general boundary value problems of elliptic type. Isz. Akad. Nauk, Math. Ser. 17, 539–562 (1953)Google Scholar
34. [Shu78]
Shubin, M.A.: Pseudodifferential Operators and Spectral Theory. Springer (2001)Google Scholar
35. [TS66]
Tikhonov, A.N., Samarskij, A.A.: Equations of Mathematical Physics. Pergamon-Press, New York (1963)
36. [Tre80]
Treves, F.: Introduction to Pseudo-Differential and Fourier Integral Operators. Plenum Publishing Co., New York (1980)
37. [Tsk94]
Tskhovrebadze, G.D.: On a multipoint boundary value problem for linear ordinary differential equations with singularities. Archivum Mathematicum, 30 (3), 171–206 (1994)
38. [Uma98]
Umarov, S.R.: Nonlocal boundary value problems for pseudo-differential and differential operator equations II. Differ. Equations, 34, 374–381 (1988)
39. [Vla79]
Vladimirov, V.S.: Generalized Functions in Mathematical Physics. Mir Publishers, Moscow (1979)Google Scholar
40. [Zh14]
Zhang, H.-E.: Multiple positive solutions of nonlinear BVPs for differential systems involving integral conditions. Boundary Value Problems, 6, 13 pp. (2014)Google Scholar

© Springer International Publishing Switzerland 2015

## Authors and Affiliations

• Sabir Umarov
• 1
1. 1.Department of MathematicsUniversity of New HavenWest HavenUSA

## Personalised recommendations

### Citechapter 