We establish the rectifiability of measures satisfying a linear PDE constraint. The obtained rectifiability dimensions are optimal for many usual PDE operators, including all first-order systems and all secondorder scalar operators. In particular, our general theorem provides a new proof of the rectifiability results for functions of bounded variations (BV) and functions of bounded deformation (BD). For divergence-free tensors we obtain refinements and new proofs of several known results on the rectifiability of varifolds and defect measures.

In this paper we investigate the "area blow-up" set of a sequence of smooth co-dimension one manifolds whose first variation with respect to an anisotropic integral is bounded. Following the ideas introduced by White in [12], we show that this set has bounded (anisotropic) mean curvature in the viscosity sense. In particular, this allows to show that the set is empty in a variety of situations. As a consequence, we show boundary curvature estimates for two dimensional stable anisotropic minimal surfaces, extending the results of [10].

We prove local boundedness of local minimizers of scalar integral functionals [Formula: see text], [Formula: see text] where the integrand satisfies [Formula: see text]-growth of the form [Formula: see text] under the optimal relation [Formula: see text].

We consider unbranched Willmore surfaces in the Euclidean space that arise as inverted complete minimal surfaces with embedded planar ends. Several statements are proven about upper and lower bounds on the Morse Index -the number of linearly independent variational directions that locally decrease the Willmore energy. We in particular compute the Index of a Willmore sphere in the three-space. This Index is m − d, where m is the number of ends of the corresponding complete minimal surface and d is the dimension of the span of the normals at the m-fold point. The dimension d is either two or three. For m = 4 we prove that d = 3. In general, we show that there is a strong connection of the Morse Index to the number of logarithmically growing Jacobi fields on the corresponding minimal surface.

Let Σ be a smooth Riemannian manifold, Γ ⊂ Σ a smooth closed oriented submanifold of codimension higher than 2 and T an integral area-minimizing current in Σ which bounds Γ. We prove that the set of regular points of T at the boundary is dense in Γ. Prior to our theorem the existence of any regular point was not known, except for some special choice of Σ and Γ. As a corollary of our theorem• we answer to a question of Almgren showing that, if Γ is connected, then T has at least one point p of multiplicity 1 2 , namely there is a neighborhood of the point p where T is a classical submanifold with boundary Γ;• we generalize Almgren's connectivity theorem showing that the support of T is always connected if Γ is connected; • we conclude a structural result on T when Γ consists of more than one connected component, generalizing a previous theorem proved by Hardt and Simon when Σ = R m+1 and T is m-dimensional. ContentsChapter 1. Introduction Chapter 2. Corollaries, open problems and plan of the paper 2.1. Indecomposable components of T 2.2. Almgren's question and proof of Theorem 1.9 2.3. Proof of Theorem 1.8 2.4. Plan of the proof of Theorem 1.6 2.5. Open problems Chapter 3. Stratification and reduction to collapsed points 3.1. First variation and monotonicity formula 3.2. Stratification 3.3. Proof of Theorem 3.8 3.4. Proof of Theorem 3.2, of Corollary 3.11 and of Corollary 3.12 Chapter 4. Regularity for Q − 1 2 Dir-minimizers 4.1. Preliminaries and proof of Theorem 4.2 4.1.1. Interpolation lemma 4.1.2. A simple measure theoretical lemma 4.1.3. Proof of Theorem 4.8: Compactness 4.1.4. Proof of Theorem 4.8: Minimality 4.2. The main frequency function estimate 4.2.1. H and D 4.2.2. Lower bound on H 4.2.3. Outer variations 4.2.4. Inner variations 4.2.5. A good function d 4.2.6. Proof of Theorem 4.15 4.3. Further consequences of the frequency function estimate 4.4. Blowup: proof of Theorem 4.5 with ϕ ≡ 0 4.5. Proof of Theorem 4.5: general case Chapter 5. First Lipschitz approximation and harmonic blow-up 5.1. Proof of Theorem 5.5 5.1.1. Artificial sheet and "bad set" 5.1.2. Lipschitz estimate 9.4. First variations 9.4.1. Outer variation 9.4.2. Inner variation 9.5. Key identities 9.6. Estimates on the error terms 9.6.1. Families of subregions 9.6.2. Lower and upper bounds in the subregions 9.6.3. Estimates on the error terms 9.7. Proof of Theorem 9.3 Chapter 10. Final blow-up argument 10.1. Asymptotics for D(r) 10.2. Vanishing of the average 10.3. Minimality and convergence in energy Bibliography Index CHAPTER 1