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Additional resources for Nonlinear Partial Differential Equations with Applications
Besides, we will conﬁne ourselves to data with polynomial-growth; p ∈ (1, +∞) will denote the growth of the leading nonlinearity a(x, u, ·) which essentially determines the setting and the other data qualiﬁcation. Also, a(x, u, ·) will be assumed to behave monotonically, cf. 65), which is related to the adjective elliptic. 91a) below implies the matrix A is positive deﬁnite, which is what is conventionally called “elliptic”, contrary to A indeﬁnite (resp. semideﬁnite) which is addressed as hyperbolic (resp.
0,a) (x) = 1 for x ∈ (0, a) and χ(0,a) (x) = 0 for x ∈ [a, 1). This collection is an uncountable subset of the unit sphere in L∞ (0, 1) and χ(0,a) − χ(0,b) L∞ (0,1) = 1 for a = b, hence L∞ (0, 1) cannot be separable. 45 Cf. 55 below. g. 26]. e. is u (l,m) a characteristic function of the interval of the type [(m − 1)2−l , m2−l ] for 1 ≤ m ≤ 2l , arranged lexicographically as (l, m) = (1, 1), (1, 2), (2, 1), (2, 2), (2, 3), (2, 4), (3, 1), . .. e. , by taking m = 1 only. 47 Obviously, existence of a common integrable minorant can weaken this assumption.
This condition is referred to as a (nonlinear) Newton boundary condition or sometimes also a Robin condition. If b = 0, it is called a Neumann boundary condition. 48) on various parts of Γ. For this, let us divide (up to a zero-measure set) the boundary Γ on two disjoint open parts ΓD and ΓN such that measn−1 Γ \ (ΓD ∪ ΓN ) = 0, and then consider so-called mixed boundary conditions u|Γ = uD ν · a(x, u, ∇u) + b(x, u) = h on ΓD , on ΓN . 47), respectively. 42 Chapter 2. 47) (resp. 49)), we will speak about a Dirichlet (resp.