Interior Estimates of Solutions of Linear Differential Inequalities
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NARY DIFFERENTIAL EQUATIONS
Interior Estimates of Solutions of Linear Differential Inequalities V. S. Klimov1∗ 1
Demidov Yaroslavl State University, Yaroslavl, 150003 Russia e-mail: ∗ [email protected]
Received October 24, 2018; revised May 14, 2020; accepted May 14, 2020
Abstract—For the solutions of the linear differential inequality L (u) ≥ 0, where L is a linear differential operator of order l defined on functions of one variable, we establish estimates of the form ku; W l (J δ )k ≤ C(δ)ku; L(J)k, where J = [a, b] ⊂ R, 0 < 3δ < b − a, J δ = [a + δ, b − δ], W l (J δ ) is the Sobolev space of l times differentiable functions, L(J) is the Lebesgue space of integrable functions, and the constant C(δ) is independent of the choice of the function u. DOI: 10.1134/S0012266120080042
INTRODUCTION Direct and inverse functional inequalities are frequently used in the theory of differential equations. A typical example of direct inequalities is given by embedding theorems, in which substantial part is about estimates for the norms of functions in terms of derivatives of these functions. Inverse functional inequalities give estimates for the norms of higher derivatives in terms of the norms of lower derivatives. Estimates of this kind only hold for functions satisfying additional conditions. In the present paper, we study the solutions of the differential inequality L (x) ≥ 0, where L is a linear differential operator of order l defined on functions of one variable. The set of solutions of this inequality is a wedge K (L ) in an appropriate Sobolev space. Recall some definitions. A closed subset K of a real normed space E is called a wedge if for any x, y ∈ K and α ≥ 0 one has x + y ∈ K and αx ∈ K. If K is a wedge, then the set K ∩ (−K) is referred to as its edge. A wedge K is called a cone if its edge consists of a single point. In Sec. 1, we consider the case in which L (x) is a locally integrable function. For the elements of the wedge K (L ) we prove an inverse inequality of the form
Z Z l Z X (i) x (t) dt ≤ M1 |x(t)| dt + M2 |x(t)| dt, δl i=0 Jδ
J
J\J δ
where J = [a, b], J δ = [a + δ, b − δ], 0 < 3δ < b − a, and M1 and M2 are some positive constants. In Sec. 2, the opposite inequalities are proved for the case in which L (x) is a nonnegative Radon measure. The concluding section deals with the discussion of the results obtained. We use the following notation: R is the field of real numbers, and N is the set of positive integers. By Rm we denote the m-dimensional real arithmetic space; the inner product of vectors m u = (up is defined by the formula (u, v) = u1 v1 + . . . + um vm ; 1 , . . . , um ) and v = (v1 , . . . , vm ) in R |u| = (u, u) is the Euclidean norm of a vector u; Rm + = {u = (u1 , . . . , um ): u1 ≥ 0, . . . , um ≥ 0} is the cone in Rm defining a semiorder on the space Rm . For x, y in Rm , we write x ≥ y if x − y ∈ Rm +. All Banach spaces are considered over the field R. If E is a Banach space and x ∈ E, then by kx; Ek we denote the norm of the element x in the space E; E ∗ is the dual space of E; σ(
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