Existence of an optimal shape of a deformable body made from a physically nonlinear material obeying a specific nonlinear generalized Hooke’s law (in fact, the so called deformation theory of plasticity is invoked in this case) is proved. Approximation of the problem by finite elements is also discussed.

Let $n\ge 2$ and $\Omega \subset {ℝ}^n$ be a bounded set. We give a Moser-type inequality for an embedding of the Orlicz-Sobolev space $W_0{L}^{\Phi }\left(\Omega \right)$, where the Young function $\Phi$ behaves like $t^n{log}^{\alpha }\left(t\right)$, $\alpha <n-1$, for $t$ large, into the Zygmund space $Z_0^{\frac{n-1-\alpha }{n}}\left(\Omega \right)$. We also study the same problem for the embedding of the generalized Lorentz-Sobolev space $W_0^m{L}^{\frac{n}{m},q}{log}^{\alpha }L\left(\Omega \right)$, $m<n$, $q\in (1,\infty ]$, $\alpha <\frac{1}{q^{\text{'}}}$, embedded into the Zygmund space $Z_0^{\frac{1}{q^{\text{'}}}-\alpha }\left(\Omega \right)$.

We consider function spaces of Besov, Triebel-Lizorkin, Bessel-potential and Sobolev type on ${ℝ}^d$, equipped with power weights $w\left(x\right)={\left|x\right|}^{\gamma }$, γ > -d. We prove two-weight Sobolev embeddings for these spaces. Moreover, we precisely characterize for which parameters the embeddings hold. The proofs are presented in such a way that they also hold for vector-valued functions.

We prove sharp embeddings of Besov spaces Bp,rσ,α with the classical smoothness σ and a logarithmic smoothness α into Lorentz-Zygmund spaces. Our results extend those with α = 0, which have been proved by D. E. Edmunds and H. Triebel. On page 88 of their paper (Math. Nachr. 207 (1999), 79-92) they have written: ?Nevertheless a direct proof, avoiding the machinery of function spaces, would be desirable.? In our paper we give such a proof even in a more general context. We cover both the sub-limiting...

Sharp estimates are obtained for the rates of blow up of the norms of embeddings of Besov spaces in Lorentz spaces as the parameters approach critical values.

We prove that the generalized Trudinger inequality for Orlicz-Sobolev spaces embedded into multiple exponential spaces implies a version of an inequality due to Brézis and Wainger.

We establish the following sharp local estimate for the family ${R_j}_{j=1}^d$ of Riesz transforms on ${ℝ}^d$. For any Borel subset A of ${ℝ}^d$ and any function $f:{ℝ}^d\to ℝ$, ${\int }_A|R_{j}f\left(x\right)|dx\le C_p{\left|\right|f\left|\right|}_{L^p\left({ℝ}^d\right)}{\left|A\right|}^{1/q}$, 1 < p < ∞. Here q = p/(p-1) is the harmonic conjugate to p, $C_p={[2^{q+2}\Gamma (q+1)/{\pi }^{q+1}{\sum }_{k=0}^{\infty }{(-1)}^k/{(2k+1)}^{q+1}]}^{1/q}$, 1 < p < 2, and $C_p={[4\Gamma (q+1)/{\pi }^q{\sum }_{k=0}^{\infty }1/{(2k+1)}^q]}^{1/q}$, 2 ≤ p < ∞. This enables us to determine the precise values of the weak-type constants for Riesz transforms for 1 < p < ∞. The proof rests on appropriate martingale inequalities, which are of independent interest.

Sharp $L^p-L^q$ estimates are obtained for averaging operators associated to hypersurfaces in $R^n$ given as graphs of homogeneous functions. An application to the regularity of an initial value problem is given.

For any locally integrable f on ℝⁿ, we consider the operators S and T which average f over balls of radius |x| and center 0 and x, respectively: $Sf\left(x\right)=1/\left|B\right(0,\left|x\right|\left)\right|{\int }_{B(0,|x\left|\right)}f\left(t\right)dt$, $Tf\left(x\right)=1/\left|B\right(x,\left|x\right|\left)\right|{\int }_{B(x,|x\left|\right)}f\left(t\right)dt$ for x ∈ ℝⁿ. The purpose of the paper is to establish sharp localized LlogL estimates for S and T. The proof rests on a corresponding one-weight estimate for a martingale maximal function, a result which is of independent interest.

Let ${\left(h_k\right)}_{k\ge 0}$ be the Haar system on [0,1]. We show that for any vectors $a_k$ from a separable Hilbert space and any ${\epsilon }_k\in [-1,1]$, k = 0,1,2,..., we have the sharp inequality $\left|\right|{\sum }_{k=0}^n{\epsilon }_k{a}_k{h}_k{\left|\right|}_{W\left(\right[0,1\left]\right)}\le 2\left|\right|{\sum }_{k=0}^n{a}_k{h}_k{\left|\right|}_{L^{\infty }\left([0,1]\right)}$, n = 0,1,2,..., where W([0,1]) is the weak-$L^{\infty }$ space introduced by Bennett, DeVore and Sharpley. The above estimate is generalized to the sharp weak-type bound ${\left|\right|Y\left|\right|}_{W\left(\Omega \right)}{\le 2\left|\right|X\left|\right|}_{L^{\infty }\left(\Omega \right)}$, where X and Y stand for -valued martingales such that Y is differentially subordinate to X. An application to harmonic functions on Euclidean domains is presented.