By Bengt O. Turesson
The e-book systematically develops the nonlinear power idea attached with the weighted Sobolev areas, the place the load often belongs to Muckenhoupt's category of Ap weights. those areas ensue as ideas areas for degenerate elliptic partial differential equations. The Sobolev house thought covers effects touching on approximation, extension, and interpolation, Sobolev and Poincaré inequalities, Maz'ya kind embedding theorems, and isoperimetric inequalities. within the bankruptcy dedicated to power conception, a number of weighted capacities are investigated. furthermore, "Kellogg lemmas" are verified for numerous thoughts of thinness. functions of power concept to weighted Sobolev areas comprise quasi continuity of Sobolev capabilities, Poincaré inequalities, and spectral synthesis theorems.
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Extra resources for Nonlinear Potential Theory and Weighted Sobolev Spaces
Sample text
6) that there is a subsequence {vn, } such that Dt~(~ * v,~) -~ v E L~(f~). 6) holds for u E W~'P(f~) n Coo(f~). If u E W~,P(fl), we choose a sequence of functions un E C°°(~) such that un --~ u in W~'P(ft). 5), applied to un, implies t h a t there is a subsequence {un~} such that Eun~ --+ Eu E W~'V(f]). e. in f~, thus completing the proof. 1. 3. Wwm'P(12) 23 An interpolation inequality With the aid of Sobolev's integral formula, we shall now prove an interpolation inequality. A consequence of this result is a Wirtinger type inequality.
P(R N) <_ Cllullw:,p(, ). 5) The constant C depends only on 12, m, N, p, and the Ap constant of w. P r o o f . First, let u • C °O( ~ ) N W ~ ,p (f}). Using a partition of unity, it is easy to -~,p W~,P(RN), see that it suffices to construct; extension operators Ej : W~ (12) --+ j = 1 , . . , M , such that E j u = u on UjNf~. Let j be fixed, and set U = Uj and C = Cj. Suppose that the length of C is 2(f and that C has its vertex at 0 (for simplicity). Let X • C~°(B26(0)) be such that X _> 0 and X = 1 on Bs(0), and let ¢ • Coo(S N-l) be such that ~ _> 0 and such that the support of the function ~, ~o(x) -- X(X)¢(x/Ixl)[x[ m-N, is included in - C O (0}.
T > O, 7-/N-I(E MEt) ----0. P r o o f . By the co-area formula, 0= IVuldx= /0 7-LN-l(ENgt)dt. 9. Since w E A1, w(x) > C f B w d y for every x E B except for x belonging to a set E of measure 0. 4. e. t _> 0. 17) is defined. Then fc w(N-1)/N dT-tN-X<- C ( IBI ~ I/N \w(B)] ~e, , thus proving what we wanted. 3. 8 with lower, (N - 1)-dimensional Minkowski content, it is possible to obtain isoperimetric inequalities that hold for arbitrary compact sets. For the definition of Minkowski content in the non-weighted case, see Federer [38, p.