zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
An extension problem related to the fractional Laplacian. (English) Zbl 1143.26002
The authors relate the fractional Laplacian of a function $f: \Bbb{R}^{n}\rightarrow \Bbb{R}$ to solutions $u:\Bbb{R}^{n}\times [0,\infty )\rightarrow \Bbb{R}$ of the extension problem $$ \left\{ \matrix u(x,0)=f(x) \\ \Delta _{x}u+\frac{a}{y}u_{y}+u_{yy}=0. \endmatrix \right. $$ It is shown that $$ \lim_{y\rightarrow 0}y^{a}u_{y}(x,y)=u_{z}(x,0)=-(-\Delta )^{s}f(x) $$ where $s=\frac{1-a}{2}$ and $z=\left( \frac{y}{1-a}\right) ^{1-a}.$ This work extends the well-known fact that the operator $(-\Delta )^{1/2}$ can be obtained from the harmonic extension problem to the upper half space as the operator that maps the Dirichlet boundary condition to the Neumann condition. Therefore, the present work generalizes this characterization to general fractional powers of the Laplacian. This is also done for other integro-differential operators and some properties of these integro-differential equations are derived.

26A33Fractional derivatives and integrals (real functions)
35J70Degenerate elliptic equations
Full Text: DOI arXiv