Український математичний вiсник Том 15 (2018), № 1, 103 – 131 Improved integrability and boundedness of solutions to some high-order variational problems Mykhailo V. Voitovych (Presented by I. I. Skrypnik) Abstract. In this article, we give a series of results on the improved integrability and boundedness of solutions to several high-order varia- tional problems with strengthened coercivity. In particular, we consider the homogeneous Dirichlet problem on the minimum of integral func- tionals and study variational inequalities with unilateral and bilateral obstacles and also with integral and gradient constraints. 2010 MSC. 35A15, 35J35, 35J87, 49J40. Key words and phrases. High-order integral functional, variational inequality, strengthened coercivity, Stampacchia’s method, integrability, boundedness. 1. Introduction Let n, m ∈ N and p ∈ R be numbers such that m  2, p> 1, and n>mp. Let Ω be a bounded open set of R n . The known counterexamples [6,18] to the nineteenth Hilbert problem demonstrate that elliptic partial differential equations of the form ∑ |α|m (−1) |α| D α A α (x,u,...,D m u)=0 in Ω (1.1) can have unbounded generalized solutions when the coefficients A α are smooth functions satisfying the ordinary growth and coercivity conditions for the Sobolev space W m,p (Ω). However, in this situation, it is possible to extract a subclass of equations of the form (1.1) whose all generalized solutions are bounded and H¨older continuous [21]. This class of equations is characterized by a strengthened coercivity condition under which the Received 14.03.2018 ISSN 1810 – 3200. c ⃝ Iнститут прикладної математики i механiки НАН України