liu.seSearch for publications in DiVA
Change search
ReferencesLink to record
Permanent link

Direct link
Sharp pointwise estimates for solutions of strongly elliptic second order systems with boundary data from L-P
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
2007 (English)In: Applicable Analysis, ISSN 0003-6811, Vol. 86, no 7, 783-805 p.Article in journal (Refereed) Published
Abstract [en]

The strongly elliptic system Aij partial derivative(2)u/partial derivative x(i)partial derivative x(j) = 0 with constant m x m matrix-valued coefficients A(ij) = A(ji) for a vector-valued functions u = (u(1),...,u(m)) in the half-space R-+(n) = {x = (x(1),..., x(n)) : x(n) > 0} as well as in a domain Omega subset of R-n with smooth boundary partial derivative Omega and compact closure Omega is considered. A representation for the sharp constant C-p in the inequality vertical bar u(x)vertical bar <= C(p)x(n)((1-n)/p) parallel to u vertical bar x(n)=0 parallel to p is obtained, where vertical bar center dot vertical bar is the length of a vector in the m-dimensional Euclidean space, x epsilon R-+(n), and parallel to center dot parallel to(p) is the L-p-norm of the modulus of an m-component vector-valued function, 1 <= p <= infinity. It is shown that lim vertical bar x - O-x vertical bar((n-1)/p) sup{vertical bar u(x)vertical bar : parallel to u vertical bar partial derivative Omega parallel to p <= 1} =C-p(O-x), x -> O-x where O-x is a point at partial derivative Omega nearest to x epsilon Omega, u is the solution of Dirichlet problem in Omega for the strongly elliptic system A(ij)partial derivative(2)u/partial derivative x(i)partial derivative x(j) = 0 with boundary data from [L-p(partial derivative Omega)](m), and C-p(O-x) is the sharp constant in the aforementioned inequality for u in the tangent space R-+(n) (O-x) to partial derivative Omega at O-x. As examples, Lame ' and Stokes systems are considered. For instance, in the case of the Stokes system, the explicit formula C-p = 2 Gamma((n + 2)/2)/pi((n+p-1)/(2p)) {Gamma((2p + n - 1)/(2p - 2))/Gamma((n + 1)p/(2p - 2))}((p-1)/p) is derived, where 1 < p < infinity.

Place, publisher, year, edition, pages
2007. Vol. 86, no 7, 783-805 p.
Keyword [en]
boundary L-P-data, pointwise estimates, strongly elliptic systems, lame and stokes systems
National Category
Engineering and Technology
URN: urn:nbn:se:liu:diva-45929DOI: 10.1080/00036810601094337OAI: diva2:266825
Available from: 2009-10-11 Created: 2009-10-11 Last updated: 2011-01-11

Open Access in DiVA

No full text

Other links

Publisher's full text

Search in DiVA

By author/editor
Maz´ya, Vladimir G.
By organisation
The Institute of TechnologyApplied Mathematics
In the same journal
Applicable Analysis
Engineering and Technology

Search outside of DiVA

GoogleGoogle Scholar
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

Altmetric score

Total: 292 hits
ReferencesLink to record
Permanent link

Direct link