## Boundary values of solutions of elliptic equations satisfying $H^{p}$ conditions

HTML articles powered by AMS MathViewer

- by Robert S. Strichartz PDF
- Trans. Amer. Math. Soc.
**176**(1973), 445-462 Request permission

## Abstract:

Let*A*be an elliptic linear partial differential operator with ${C^\infty }$ coefficients on a manifold ${\mathbf {\Omega }}$ with boundary ${\mathbf {\Gamma }}$. We study solutions of $Au = \sigma$ which satisfy the ${H^p}$ condition that ${\sup _{0 < t < 1}}{\left \| {u( \cdot ,t)} \right \|_p} < \infty$, where we have chosen coordinates in a neighborhood of ${\mathbf {\Gamma }}$ of the form ${\mathbf {\Gamma }} \times [0,1]$ with ${\mathbf {\Gamma }}$ identified with $t = 0$. If

*A*has a well-posed Dirichlet problem such solutions may be characterized in terms of the Dirichlet data $u( \cdot ,0) = {f_0},{(\partial /\partial t)^j}u( \cdot ,0) = {f_j},j = 1, \cdots ,m - 1$ as follows: ${f_0} \in {L^p}$ (or $\mathfrak {M}$ if $p = 1$) and ${f_j} \in {\mathbf {\Lambda }}( - j;p,\infty ),j = 1, \cdots ,m$ . Here ${\mathbf {\Lambda }}$ denotes the Besov spaces in Taibleson’s notation. If $m = 1$ then

*u*has nontangential limits almost everywhere.

## References

- S. Agmon, A. Douglis, and L. Nirenberg,
*Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I*, Comm. Pure Appl. Math.**12**(1959), 623–727. MR**125307**, DOI 10.1002/cpa.3160120405 - A. Benedek, A.-P. Calderón, and R. Panzone,
*Convolution operators on Banach space valued functions*, Proc. Nat. Acad. Sci. U.S.A.**48**(1962), 356–365. MR**133653**, DOI 10.1073/pnas.48.3.356
A. P. Calderón, - J. R. Hattemer,
*Boundary behavior of temperatures. I*, Studia Math.**25**(1964/65), 111–155. MR**181838**, DOI 10.4064/sm-25-1-111-155 - Richard A. Hunt and Richard L. Wheeden,
*On the boundary values of harmonic functions*, Trans. Amer. Math. Soc.**132**(1968), 307–322. MR**226044**, DOI 10.1090/S0002-9947-1968-0226044-7 - Jaak Peetre,
*Sur les espaces de Besov*, C. R. Acad. Sci. Paris Sér. A-B**264**(1967), A281–A283 (French). MR**218887** - John C. Polking,
*Boundary value problems for parabolic systems of partial differential equations*, Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966) Amer. Math. Soc., Providence, R.I., 1967, pp. 243–274. MR**0378000** - R. T. Seeley,
*Singular integrals and boundary value problems*, Amer. J. Math.**88**(1966), 781–809. MR**209915**, DOI 10.2307/2373078 - R. Seeley,
*Topics in pseudo-differential operators*, Pseudo-Diff. Operators (C.I.M.E., Stresa, 1968) Edizioni Cremonese, Rome, 1969, pp. 167–305. MR**0259335** - Elias M. Stein,
*Singular integrals and differentiability properties of functions*, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR**0290095** - E. M. Stein,
*The characterization of functions arising as potentials. II*, Bull. Amer. Math. Soc.**68**(1962), 577–582. MR**142980**, DOI 10.1090/S0002-9904-1962-10856-8
M. H. Taibleson, - Kjell-Ove Widman,
*On the boundary behavior of solutions to a class of elliptic partial differential equations*, Ark. Mat.**6**(1967), 485–533 (1967). MR**219875**, DOI 10.1007/BF02591926 - A. Zygmund,
*Trigonometric series: Vols. I, II*, Cambridge University Press, London-New York, 1968. Second edition, reprinted with corrections and some additions. MR**0236587**

*A priori estimates for singular integral operators*, Pseudo-Differential Operators (C.I.M.E., Stresa, 1968), Edizioni Cremonese, Rome, 1969, pp. 85-141. MR

**41**#872.

*On the theory of Lipschitz spaces of distrubutions on Euclidean n-space*. I, II, J. Math. Mech.

**13**(1964), 407-479; ibid. 14 (1965), 821-839. MR

**29**#462; MR

**31**#5087.

## Additional Information

- © Copyright 1973 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**176**(1973), 445-462 - MSC: Primary 35J67
- DOI: https://doi.org/10.1090/S0002-9947-1973-0320525-5
- MathSciNet review: 0320525