Limiting Sobolev inequalities for vector fields and canceling linear differential operators

作者：Van Schaftingen Jean^*

来源：Journal of the European Mathematical Society, 2013, 15(3): 877-921.

DOI：10.4171/JEMS/380

• Sobolev embedding
• overdetermined elliptic operator
• compatibility conditions
• homogeneous differential operator
• canceling operator
• cocanceling operator
• exterior derivative
• symmetric derivative
• homogeneous Triebel-Lizorkin space
• homogeneous Besov space
• Lorentz space
• homogeneous fractional Sobolev-Slobodetskii space
• Korn-Sobolev inequality
• Hodge inequality
• Saint-Venant compatibility conditions

The estimate %26lt;br%26gt;parallel to D(k-1)u parallel to(Ln/(n-1)) %26lt;= parallel to A(D)u parallel to(L1) %26lt;br%26gt;is shown to hold if and only if A(D) is elliptic and canceling. Here A(D) is a homogeneous linear differential operator A(D) of order k on R-n from a vector space V to a vector space E. The operator A(D) is defined to be canceling if %26lt;br%26gt;(xi is an element of Rn\{0})boolean AND A(xi)[V] = {0}. %26lt;br%26gt;This result implies in particular the classical Gagliardo-Nirenberg-Sobolev inequality, the Korn-Sobolev inequality and Hodge-Sobolev estimates for differential forms due to J. Bourgain and H. Brezis. In the proof, the class of cocanceling homogeneous linear differential operator L(D) of order k on R-n from a vector space E to a vector space F is introduced. It is proved that L(D) is cocanceling if and only if for every f is an element of L-1 (R-n; E) such that L(D)f = 0, one has f is an element of (W) over dot(-1,n/(n-1)) (R-n; E). The results extend to fractional and Lorentz spaces and can be strengthened using some tools due to J. Bourgain and H. Brezis.

• 出版日期2013