scientific article; zbMATH DE number 3554969
From MaRDI portal
Publication:4129368
zbMath0356.49008MaRDI QIDQ4129368
Luciano Modica, Stefano Mortola
Publication date: 1977
Title: zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Methods involving semicontinuity and convergence; relaxation (49J45) Existence theories for problems in abstract spaces (49J27)
Related Items
Large data limit for a phase transition model with the p-Laplacian on point clouds ⋮ A consistent approximation of the total perimeter functional for topology optimization algorithms ⋮ Analysis of gradient flow of a regularized Mumford-Shah functional for image segmentation and image inpainting ⋮ Multiple Delaunay ends solutions of the Cahn-Hilliard equation ⋮ Unnamed Item ⋮ A Diffuse Interface Model for Cell Blebbing Including Membrane-Cortex Coupling with Linker Dynamics ⋮ Topology optimization for quasistatic elastoplasticity ⋮ A multiphase Cahn–Hilliard system with mobilities and the numerical simulation of dewetting ⋮ Gaussian Approximations for Transition Paths in Brownian Dynamics ⋮ Truncated nonsmooth Newton multigrid for phase-field brittle-fracture problems, with analysis ⋮ Classification and image processing with a semi‐discrete scheme for fidelity forced Allen–Cahn on graphs ⋮ Sharp interface limit of the Cahn–Hilliard reaction model for lithium-ion batteries ⋮ A relation of the Allen-Cahn equations and the Euler equations and applications of the equipartition ⋮ The Yang-Mills-Higgs functional on complex line bundles: \(\Gamma\)-convergence and the London equation ⋮ Phase separation on varying surfaces and convergence of diffuse interface approximations ⋮ Homogenization and phase separation with space dependent wells: the subcritical case ⋮ Discrete Approximation of the Griffith Functional by Adaptive Finite Elements ⋮ Applications of PDEs inpainting to magnetic particle imaging and corneal topography ⋮ Geometric aspects of shape optimization ⋮ “Gradient-free” diffuse approximations of the Willmore functional and Willmore flow ⋮ Two-Scale Elastic Shape Optimization for Additive Manufacturing ⋮ Material optimization for nonlinearly elastic planar beams ⋮ Interfacial energy as a selection mechanism for minimizing gradient Young measures in a one-dimensional model problem ⋮ Variational approximation of size-mass energies fork-dimensional currents ⋮ A gradient system with a wiggly energy and relaxed EDP-convergence ⋮ Locations of interior transition layers to inhomogeneous transition problems in higher -dimensional domains ⋮ Joint Reconstruction-Segmentation on Graphs ⋮ Analysis of a combined filtered/phase-field approach to topology optimization in elasticity ⋮ Multiplicity of solutions to the multiphasic Allen-Cahn-Hilliard system with a small volume constraint on closed parallelizable manifolds ⋮ Mass concentration in rescaled first order integral functionals ⋮ On the boundary regularity of phase-fields for Willmore's energy ⋮ Mean curvature flow and low energy solutions of the parabolic Allen-Cahn equation on the three-sphere ⋮ \(\Gamma\)-convergence and stochastic homogenisation of singularly-perturbed elliptic functionals ⋮ On the Reconstruction of Cavities in a Nonlinear Model Arising from Cardiac Electrophysiology ⋮ Approximation of multiphase mean curvature flows with arbitrary nonnegative mobilities ⋮ Entire solutions to 4 dimensional Ginzburg-Landau equations and codimension 2 minimal submanifolds ⋮ Connections between Operator-Splitting Methods and Deep Neural Networks with Applications in Image Segmentation ⋮ Super-resolution surface reconstruction from few low-resolution slices ⋮ Γ-convergence and stochastic homogenisation of phase-transition functionals ⋮ The Alt–Phillips functional for negative powers ⋮ Convergence of the self‐dual U(1)‐Yang–Mills–Higgs energies to the (n−2)$(n-2)$‐area functional ⋮ Two-dimensional ferronematics, canonical harmonic maps and minimal connections ⋮ Multiplicity‐1 minmax minimal hypersurfaces in manifolds with positive Ricci curvature ⋮ Fractional elliptic problems on Lipschitz domains: regularity and approximation ⋮ Weak solutions of Mullins-Sekerka flow as a Hilbert space gradient flow ⋮ Partial differential equations from theory to applications: dedicated to Alberto Farina, on the occasion of his 50th birthday ⋮ Uniform density estimates and \(\Gamma \)-convergence for the Alt-Phillips functional of negative powers ⋮ Gradient theory of domain walls in thin, nematic liquid crystals films ⋮ Symmetry results for the solutions of a partial differential equation arising in water waves ⋮ A quasi-optimal error estimate for a discrete singularly perturbed approximation to the prescribed curvature problem ⋮ The variational formulation of brittle fracture: numerical implementation and extensions ⋮ Topology Optimization for Incremental Elastoplasticity: A Phase-Field Approach ⋮ High Accuracy Benchmark Problems for Allen-Cahn and Cahn-Hilliard Dynamics ⋮ Motion of a Droplet for the Stochastic Mass-Conserving Allen--Cahn Equation ⋮ Second-order \(\Gamma\)-limit for the Cahn-Hilliard functional ⋮ One-dimensional solutions of non-local Allen-Cahn-type equations with rough kernels ⋮ Minimal surfaces and free boundaries: Recent developments ⋮ Adaptive reconstruction for electrical impedance tomography with a piecewise constant conductivity ⋮ Phase-field modelling and computing for a large number of phases ⋮ Three-Dimensional Volume Reconstruction Using Two-Dimensional Parallel Slices ⋮ Asymptotic behavior of the W1∕q,q-norm of mollified BV functions and applications to singular perturbation problems ⋮ On material optimisation for nonlinearly elastic plates and shells ⋮ An Analysis of Chevrons in Thin Liquid Crystal Cells ⋮ Phase Field Methods for Binary Recovery ⋮ The Hele–Shaw flow as the sharp interface limit of the Cahn–Hilliard equation with disparate mobilities ⋮ Two-Phase Segmentation for Intensity Inhomogeneous Images by the Allen--Cahn Local Binary Fitting Model ⋮ Phase-field methods for spectral shape and topology optimization ⋮ A sharp-interface limit for a two-well problem in geometrically linear elasticity ⋮ Droplet solutions in the diblock copolymer problem with skewed monomer composition ⋮ On the convergence of solutions of some evolution differential equations ⋮ A maximum principle for the system \(\Delta u- \nabla W(u)=0\) ⋮ On De Giorgi's conjecture: recent progress and open problems ⋮ Simultaneous reconstruction and segmentation with the Mumford-Shah functional for electron tomography ⋮ Motion of concentration sets in Ginzburg-Landau equations. ⋮ Critical points via \(\Gamma \)-convergence: general theory and applications ⋮ Saddle-shaped solutions of bistable diffusion equations in all of \(\mathbb R^{2m}\) ⋮ Gradient-flow techniques for the analysis of numerical schemes for multi-phase mean-curvature flow ⋮ A metric-based approach to multiphase mean curvature flows with mobilities ⋮ The effect of a singular perturbation on nonconvex variational problems ⋮ Geometrically constrained walls ⋮ Convergence of the thresholding scheme for multi-phase mean-curvature flow ⋮ Two-phase flow with mass density contrast: stable schemes for a thermodynamic consistent and frame-indifferent diffuse-interface model ⋮ Computationally efficient approach for the minimization of volume constrained vector-valued Ginzburg-Landau energy functional ⋮ Long-time behavior of a nonlocal Cahn-Hilliard equation with reaction ⋮ On a singular perturbation problem related to optimal lifting in BV-space ⋮ Multimaterial topology optimization by volume constrained Allen-Cahn system and regularized projected steepest descent method ⋮ Kinks in two-phase lipid bilayer membranes ⋮ A Gamma-convergence argument for the blow-up of a non-local semilinear parabolic equation with Neumann boundary conditions ⋮ Image segmentation using some piecewise constant level set methods with MBO type of projection ⋮ Global dynamics of boundary droplets ⋮ A two-gradient approach for phase transitions in thin films ⋮ Density estimates for a variational model driven by the Gagliardo norm ⋮ Regularization of discrete contour by Willmore energy ⋮ On the \(\gamma\)-limit of singular perturbation problems with optimal profiles which are not one-dimensional. II: The lower bound ⋮ \(\Gamma \)-convergence of variational functionals with boundary terms in Stein manifolds ⋮ A two scale \(\Gamma \)-convergence approach for random non-convex homogenization ⋮ A survey on the Blake-Zisserman functional ⋮ Regularity of the Eikonal equation with two vanishing entropies ⋮ A geometric Ginzburg-Landau problem ⋮ A novel homogenization method for phase field approaches based on partial rank-one relaxation ⋮ Convergence of phase-field free energy and boundary force for molecular solvation ⋮ Actuation of thin nematic elastomer sheets with controlled heterogeneity ⋮ Homogenization of Cahn-Hilliard-type equations via evolutionary \(\Gamma\)-convergence ⋮ Singular perturbations of variational problems arising from a two-phase transition model ⋮ A new stochastic variational PDE model for soft Mumford-Shah segmentation ⋮ Variational problems in SBV and image segmentation ⋮ A zigzag pattern in micromagnetics ⋮ Some properties of Gamma-limits of integral functionals ⋮ Approximation and relaxation of perimeter in the Wiener space ⋮ Asymptotics for the fractional Allen-Cahn equation and stationary nonlocal minimal surfaces ⋮ The role of Gauss curvature in a membrane phase separation problem ⋮ Asymptotic behavior of the Landau-Lifshitz model of ferromagnetism ⋮ A fractal shape optimization problem in branched transport ⋮ Minimal partitions and image classification using a gradient-free perimeter approximation ⋮ On De Giorgi's conjecture in dimension \(N\geq 9\) ⋮ A Gamma-convergence approach to the Cahn-Hilliard equation ⋮ A weighted gradient theory of phase transitions with a possibly singular and degenerate spatial inhomogeneity ⋮ Second order asymptotic development for the anisotropic Cahn-Hilliard functional ⋮ Gamma limit of the nonself-dual Chern-Simons-Higgs energy ⋮ Optimal profiles in a phase-transition model with a saturating flux ⋮ A phase-field approximation of the Steiner problem in dimension two ⋮ Phase-field approximations of the Willmore functional and flow ⋮ New asymptotic analysis method for phase field models in moving boundary problem with surface tension ⋮ Isogeometric analysis for topology optimization with a phase field model ⋮ Large deviation principle for a stochastic Allen-Cahn equation ⋮ Some results on surface measures in calculus of variations ⋮ Clifford Tori and the singularly perturbed Cahn-Hilliard equation ⋮ Periodic solutions to a Cahn-Hilliard-Willmore equation in the plane ⋮ A necessary condition for lower semicontinuity of line energies ⋮ Nonlocal phase transitions in homogeneous and periodic media ⋮ \(\Gamma\)-limits of singular perturbation problems involving energies with non-local terms ⋮ Uniqueness and stability of the saddle-shaped solution to the fractional Allen-Cahn equation ⋮ Phase transitions with a minimal number of jumps in the singular limits of higher order theories ⋮ Ambrosio-Tortorelli approximation of quasi-static evolution of brittle fractures ⋮ Reconstruction of material losses by perimeter penalization and phase-field methods ⋮ Semilinear integro-differential equations. I. Odd solutions with respect to the Simons cone ⋮ Soliton-stripe patterns of a functional with an attractive-repulsive-attractive interaction ⋮ Graph clustering, variational image segmentation methods and Hough transform scale detection for object measurement in images ⋮ A one-dimensional variational problem for cholesteric liquid crystals with disparate elastic constants ⋮ Regional control for spatially structured mosquito borne epidemics. Malaria as a working example ⋮ Regional control for spatially structured mosquito borne epidemics. II: Computational issues ⋮ Chirality transitions in frustrated ferromagnetic spin chains: a link with the gradient theory of phase transitions ⋮ Multiphase mean curvature flows with high mobility contrasts: a phase-field approach, with applications to nanowires ⋮ Derivation of a linearised elasticity model from singularly perturbed multiwell energy functionals ⋮ A Modica-Mortola approximation for branched transport ⋮ Asymptotic behaviour of time stepping methods for phase field models ⋮ A fractional isoperimetric problem in the Wiener space ⋮ A variational singular perturbation problem motivated by Ericksen's model for nematic liquid crystals ⋮ Variational approximation of interface energies and applications ⋮ Compactness and sharp lower bound for a 2D smectics model ⋮ Travelling wave solutions for the discrete sine-Gordon equation with nonlinear pair interaction ⋮ Phase-field approximation of functionals defined on piecewise-rigid maps ⋮ Sharp interface limit of a multi-phase transitions model under nonisothermal conditions ⋮ One-dimensionality of the minimizers in the large volume limit for a diffuse interface attractive/repulsive model in general dimension ⋮ The gradient theory of phase transitions and the minimal interface criterion ⋮ The quasistationary phase field equations with Neumann boundary conditions ⋮ Numerical experiments in revisited brittle fracture ⋮ The geometric structure of interfaces and free boundaries ⋮ Rigidity of a non-elliptic differential inclusion related to the Aviles-giga conjecture ⋮ Second order threshold dynamics schemes for two phase motion by mean curvature ⋮ Existence of weak solutions for a non-classical sharp interface model for a two-phase flow of viscous, incompressible fluids ⋮ The characterization of Riemannian metric arising from phase transition problems ⋮ Codimension one minimal cycles with coefficients in \(Z\) or \(Z_p\), and variational functionals on fibered spaces ⋮ On the variation of the fractional mean curvature under the effect of \(C^{1, \alpha}\) perturbations ⋮ Reorientation of smectic a liquid crystals by magnetic fields ⋮ Continuum limit of total variation on point clouds ⋮ Singular perturbation by bending for an adhesive obstacle problem ⋮ Longest minimal length partitions ⋮ A rigorous sharp interface limit of a diffuse interface model related to tumor growth ⋮ Generation of tubular and membranous shape textures with curvature functionals ⋮ Anisotropic surface tensions for phase transitions in periodic media ⋮ Asymptotic development by \(\Gamma{}\)-convergence ⋮ On the total variation of the gradient of a function ⋮ Nonconvex variational problems with anisotropic perturbations ⋮ Higher-order phase transitions with line-tension effect ⋮ Singular Perturbation Problems Involving Curvature ⋮ Fracture models for elasto-plastic materials as limits of gradient damage models coupled with plasticity: the antiplane case ⋮ A variational principle for pulsating standing waves and an Einstein relation in the sharp interface limit ⋮ A counterexample to a conjecture by De Giorgi in large dimensions ⋮ A variational study of some hadron bag models ⋮ On global minimizers for a variational problem with long-range interactions ⋮ Second-Order Edge-Penalization in the Ambrosio--Tortorelli functional ⋮ Asymptotic behavior of the interface for entire vector minimizers in phase transitions ⋮ A Variational Approach to the Inverse Photolithography Problem ⋮ On the approximation of the elastica functional in radial symmetry ⋮ Mean curvature properties for \(p\)-Laplace phase transitions ⋮ Regularity of gradient vector fields giving rise to finite Caccioppoli partitions ⋮ Variational Approximation of Functionals Defined on 1-dimensional Connected Sets: The Planar Case ⋮ Density Estimates for Degenerate Double-Well Potentials ⋮ Analysis of the Diffuse Domain Approach for a Bulk-Surface Coupled PDE System ⋮ Numerical investigations of the role of curvature in strong segregation problems on a given surface ⋮ Surface tension and \(\Gamma\)-convergence of Van der Waals-Cahn-Hilliard phase transitions in stationary ergodic media ⋮ Minimal submanifolds from the abelian Higgs model ⋮ On \(\Gamma\)-convergence of a variational model for Lithium-ion batteries ⋮ Partial differential equations and variational methods for geometric processing of images ⋮ Optimal design of multiphase composites under elastodynamic loading ⋮ K-ended \(O(m) \times O(n)\) invariant solutions to the Allen-Cahn equation with infinite Morse index ⋮ Convergence rates for the Allen-Cahn equation with boundary contact energy: the non-perturbative regime ⋮ A two-phase two-fluxes degenerate Cahn-Hilliard model as constrained Wasserstein gradient flow ⋮ A finer singular limit of a single-well Modica-Mortola functional and its applications to the Kobayashi-Warren-Carter energy ⋮ Weak Lower Semicontinuity of Integral Functionals and Applications ⋮ Singular perturbations of variational problems of vector valued functions ⋮ On the Łojasiewicz-Simon gradient inequality on submanifolds ⋮ Exponentially slow motion of interface layers for the one-dimensional Allen-Cahn equation with nonlinear phase-dependent diffusivity ⋮ Numerical minimization of a second-order functional for image segmentation ⋮ Overhang penalization in additive manufacturing via phase field structural topology optimization with anisotropic energies ⋮ A posteriori error control for the binary Mumford-Shah model ⋮ Compactness estimates for minimizers of the Alt-Phillips functional of negative exponents ⋮ On a conjecture of De Giorgi about the phase-field approximation of the Willmore functional ⋮ Symmetry and asymmetry of components for elliptic Gross-Pitaevskii system ⋮ Lamellar phase solutions for diblock copolymers with nonlocal diffusions ⋮ Double tori solution to an equation of mean curvature and Newtonian potential ⋮ A diffused interface with the advection term in a Sobolev space ⋮ The double-power nonlinear Schrödinger equation and its generalizations: uniqueness, non-degeneracy and applications ⋮ Equilibrium for multiphase solids with Eulerian interfaces ⋮ Long-range Phase Coexistence Models: Recent Progress on the Fractional Allen-Cahn Equation ⋮ Quantitative analysis of finite-difference approximations of free-discontinuity problems ⋮ A phase-field approach to variational hierarchical surface segmentation ⋮ Sharp Interface Limits for Diffuse Interface Models for Two-Phase Flows of Viscous Incompressible Fluids ⋮ Non-existence of patterns and gradient estimates in semilinear elliptic equations with Neumann boundary conditions ⋮ Shape and topology optimization ⋮ Cahn-Hilliard phase field theory coupled to mechanics: fundamentals, numerical implementation and application to topology optimization ⋮ Anisotropic free-discontinuity functionals as the Γ-limit of second-order elliptic functionals ⋮ Approximation of Partitions of Least Perimeter by Γ-Convergence: Around Kelvin’s Conjecture ⋮ Convergence of phase-field approximations to the Gibbs-Thomson law ⋮ On variational dimension reduction in structure mechanics ⋮ Symmetry in nonlinear PDEs: some open problems ⋮ Front propagation in geometric and phase field models of stratified media ⋮ Deterministic equivalent for the Allen-Cahn energy of a scaling law in the Ising model ⋮ Finite element discretizations of nonlocal minimal graphs: Convergence ⋮ Uniform estimates for the parabolic Ginzburg–Landau equation ⋮ Variational approximation of anisotropic functionals on partitions ⋮ Towards a counter-example to a conjecture of De Giorgi in high dimensions ⋮ Design-dependent loads in topology optimization ⋮ Local and Nonlocal Continuum Limits of Ising-Type Energies for Spin Systems ⋮ Volume Reconstruction from Slices ⋮ Qualitative and Numerical Analysis of a Spectral Problem with Perimeter Constraint ⋮ The \(\varGamma\)-limit of the two-dimensional Ohta-Kawasaki energy. I: Droplet density ⋮ On stable solutions for boundary reactions: a De Giorgi-type result in dimension \(4 + 1\) ⋮ Phase field approximations of branched transportation problems ⋮ Diffuse-Interface Approximations of Osmosis Free Boundary Problems ⋮ Two-well rigidity and multidimensional sharp-interface limits for solid-solid phase transitions ⋮ Some energy estimates for stable solutions to fractional Allen-Cahn equations ⋮ Some basic facts on the system $\Delta u - W_{u} (u) = 0$ ⋮ Sharp Interface Limit for a Phase Field Model in Structural Optimization ⋮ Phase field approach to optimal packing problems and related Cheeger clusters ⋮ Radial and cylindrical symmetry of solutions to the Cahn-Hilliard equation ⋮ Elastic curves and phase transitions ⋮ Efficient variational constitutive updates for Allen-Cahn-type phase field theory coupled to continuum mechanics ⋮ A phase-field approach to Eulerian interfacial energies ⋮ Asymptotic behavior of Allen-Cahn-type energies and Neumann eigenvalues via inner variations ⋮ Wetting on rough surfaces and contact angle hysteresis: numerical experiments based on a phase field model ⋮ Variational approximation for detecting point-like target problems ⋮ Unnamed Item ⋮ Interaction between oscillations and singular perturbations in a one-dimensional phase-field model ⋮ Gradient flows of non convex functionals in Hilbert spaces and applications ⋮ Linear and nonlinear fractional elliptic problems ⋮ Learning phase field mean curvature flows with neural networks ⋮ A new diffuse-interface approximation of the Willmore flow ⋮ One-dimensionality of the minimizers for a diffuse interface generalized antiferromagnetic model in general dimension ⋮ Convergence to equilibrium for time and space discretizations of the Cahn-Hilliard equation ⋮ Threshold dynamics for the piecewise constant Mumford-Shah functional ⋮ Asymptotic analysis of second order nonlocal Cahn-Hilliard-type functionals ⋮ Entire solutions of semilinear elliptic equations in ℝ³ and a conjecture of De Giorgi ⋮ Introduction: Big data and partial differential equations ⋮ Frustrated ferromagnetic spin chains: a variational approach to chirality transitions ⋮ \(\Gamma\)-convergence approximation of fracture and cavitation in nonlinear elasticity