Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. (AM-105)

Lp-Mean coercivity, regularity and relaxation in the calculus of variations

“…Under this condition, some of the properties concerning the relaxation (f p ) qc turns out independent of the power p. For example, it has been proved in Yan [35] that [12,38]), where Z[g] denotes the zero set of function g. This result will be partially recovered later from our main results in which we replace (1.4) by a much weaker condition known as the L p -mean coercivity [14,15,18]. We say that I p (u; D) or simply f satisÿes the L p -mean…”

“…This can be considered as a global version of the aforementioned results on the higher integrability of energy minimizers (see, e.g., [13][14][15]17,20,24]). …”

“…The proof uses standard techniques of Caccioppoli-type estimates [14,15,20,23,24], but requires a careful treatment since the integrand f p does not satisfy the usual growth conditions, so we present the detail here; see also Yan and Zhou [37]. Let (V A ; ) be deÿned as before, and let c 0 ; c 1 ; : : : denote the constants depending only on p and f: Given Let w = Á + (1 − Á)b j and = b j − w; where ∈ R n is a constant to be chosen later.…”

“…This type of higher integrability results, pioneered by Gehring's celebrated work [13], has been well-known for the energy minimizers of variational integrals under certain pointwise growth conditions (see [14,15,20,24,23]). Indeed, by adapting the Caccioppoli-type estimates as in Meyers and Elcrat [24] and Giaquinta and Giusti [15], Theorem 2.1 is proved by the well-known technique of reverse H older inequalities of Gehring [13].…”

Weak Estimates of Singular Integrals with Variable Kernel and Fractional Differentiation on Morrey-Herz Spaces

“…The ideas of Morrey [11] were further developed by Campanato [13]. A more systematic study of these (and even more general) spaces, we refer the readers to see [12,[14][15][16][17][18][19][20][21]. In 1964, Beurling [22] first introduced some fundamental forms of Herz spaces to study convolution algebras.…”