A lower bound on the energy of travelling waves of fixed speed for the Gross-Pitaevskii equation

In this paper we consider the Gross-Pitaevskii equation iu _t = Δu + u(1 − |u|²), where u is a complex-valued function defined on \Bbb R^N×\Bbb R{\Bbb R}^N\times{\Bbb R} , N ≥ 2, and in particular the travelling waves, i.e., the solutions of the form u(x, t) = ν(x ₁ − ct, x ₂, …, x _N ), where c ? \Bbb Rc\in{\Bbb R} is the speed. We prove for c fixed the existence of a lower bound on the energy of any non-constant travelling wave. This bound provides a non-existence result for non-constant travelling waves of fixed speed having small energy.     

On Nonexistence of type II blowup for a supercritical nonlinear heat equation

In this paper we study blowup of radially symmetric solutions of the nonlinear heat equation u_t = Δu + |u|^p?1u either on ?^N or on a finite ball under the Dirichlet boundary conditions. We assume that the exponent p is supercritical in the Sobolev sense, that is, We prove that if p_s < p < p^*, then blowup is always of type I, where p^* is a certain (explicitly given) positive number. More precisely, the rate of blowup in the L^∞ norm is always the same as that for the corresponding ODE dv/dt = |v|^p?1v. Because it is known that “type II” blowup (or, equivalently, “fast blowup”) can occur if p > p^*, the above range of exponent p is optimal. We will also derive various fundamental estimates for blowup that hold for any p > p_s and regardless of type of blowup. Among other things we classify local profiles of type I and type II blowups in the rescaled coordinates. We then establish useful estimates for the so‐called incomplete blowup, which reveal that incomplete blowup solutions belong to nice function spaces even after the blowup time. © 2004 Wiley Periodicals, Inc.     

Continuation of blowup solutions of nonlinear heat equations in several space dimensions

The possible continuation of solutions of the nonlinear heat equation in R^N × R₊ u_t = Δu^m + u^p with m > 0, p > 1, after the blowup time is studied and the different continuation modes are discussed in terms of the exponents m and p. Thus, for m + p ≤ 2 we find a phenomenon of nontrivial continuation where the region {x : u(x, t) = ∞} is bounded and propagates with finite speed. This we call incomplete blowup. For N ≥ 3 and p > m(N + 2)/(N − 2) we find solutions that blow up at finite t = T and then become bounded again for t > T. Otherwise, we find that blowup is complete for a wide class of initial data. In the analysis of the behavior for large p, a list of critical exponents appears whose role is described. We also discuss a number of related problems and equations. We apply the same technique of analysis to the problem of continuation after the onset of extinction, for example, for the equation u_t = Δu^m − u^p, m > 0. We find that no continuation exists if p + m ≤ 0 (complete extinction), and there exists a nontrivial continuation if p + m > 0 (incomplete extinction). © 1997 John Wiley & Sons, Inc.     

Lp theory for the multidimensional aggregation equation

We consider well‐posedness of the aggregation equation ∂_tu + div(uv) = 0, v = −▿K * u with initial data in \input amssym ${\cal P}_2 {\rm (\Bbb R}^d {\rm )} \cap L^p ({\Bbb R}^d )$ in dimensions 2 and higher. We consider radially symmetric kernels where the singularity at the origin is of order |x|^α, α > 2 − d, and prove local well‐posedness in \input amssym ${\cal P}_2 { (\Bbb R}^d {\rm )} \cap L^p ({\Bbb R}^d )$ for sufficiently large p < p_s. In the special case of K(x) = |x|, the exponent p_s = d/(d = 1) is sharp for local well‐posedness in that solutions can instantaneously concentrate mass for initial data in \input amssym ${\cal P}_2 { (\Bbb R}^d {\rm )} \cap L^p ({\Bbb R}^d )$ with p < p_s. We also give an Osgood condition on the potential K(x) that guarantees global existence and uniqueness in \input amssym ${\cal P}_2 { (\Bbb R}^d {\rm )} \cap L^p ({\Bbb R}^d )$ . © 2010 Wiley Periodicals, Inc.     

A lower bound on the energy of travelling waves of fixed speed for the Gross-Pitaevskii equation

In this paper we consider the Gross-Pitaevskii equation iu _t = Δu + u(1 − |u|²), where u is a complex-valued function defined on , N ≥ 2, and in particular the travelling waves, i.e., the solutions of the form u(x, t) = ν(x ₁ − ct, x ₂, …, x _N ), where is the speed. We prove for c fixed the existence of a lower bound on the energy of any non-constant travelling wave. This bound provides a non-existence result for non-constant travelling waves of fixed speed having small energy.     

Decay of mass for nonlinear equation with fractional Laplacian

The large time behavior of non-negative solutions to the reaction–diffusion equation ?_t u=-(-D)^a/2u - u^p{\partial_t u=-(-\Delta)^{\alpha/2}u - u^p}, ${(\alpha\in(0,2], \;p > 1)}${(\alpha\in(0,2], \;p > 1)} posed on \mathbbR^N{\mathbb{R}^N} and supplemented with an integrable initial condition is studied. We show that the anomalous diffusion term determines the large time asymptotics for p > 1 + α/N, while nonlinear effects win if p ≤ 1 + α/N.     

On the fragmentation of a torus by random walk

We consider a simple random walk on a discrete torus \input amssym $({\Bbb Z}/N{\Bbb Z})^d$ with dimension d ≥ 3 and large side length N. For a fixed constant u ≥ 0, we study the percolative properties of the vacant set, consisting of the set of vertices not visited by the random walk in its first [uN^d] steps. We prove the existence of two distinct phases of the vacant set in the following sense: If u > 0 is chosen large enough, all components of the vacant set contain no more than (log N)^λ(u) vertices with high probability as N tends to infinity. On the other hand, for small u > 0, there exists a macroscopic component of the vacant set occupying a nondegenerate fraction of the total volume N^d. In dimensions d ≥ 5, we additionally prove that this macroscopic component is unique by showing that all other components have volumes of order at most (log N)^λ(u). Our results thus solve open problems posed by Benjamini and Sznitman, who studied the small u regime in high dimension. The proofs are based on a coupling of the random walk with random interlacements on \input amssym ${\Bbb Z}^d$ . Among other techniques, the construction of this coupling employs a refined use of discrete potential theory. By itself, this coupling strengthens a result by Windisch. © 2011 Wiley Periodicals, Inc.     

Multiplicity of solutions for a fourth order equation with power-type nonlinearity

Let B be the unit ball in ${\mathbb{R}^N}Let B be the unit ball in \mathbbR^N{\mathbb{R}^N}, N ≥ 3 and n be the exterior unit normal vector on the boundary. We consider radial solutions to

D2 u = l(1+ sign(p)u)p     in  B,     u = 0,     \frac?u?n = 0     on  ?B\Delta^2 u = \lambda(1+ {\rm sign}(p)u)^{p} \quad {\rm in} \, B, \quad u = 0, \quad \frac{\partial{u}}{\partial{n}} = 0 \quad {\rm on} \, \partial B      9. On products of k atoms      Weidong Gao  Alfred Geroldinger 《Monatshefte für Mathematik》2009,79(1):141-157 Let H be an atomic monoid. For k ? \Bbb Nk \in {\Bbb N} let Vk (H){\cal V}_k (H) denote the set of all m ? \Bbb Nm \in {\Bbb N} with the following property: There exist atoms (irreducible elements) u 1, …, u k , v 1, …, v m ∈ H with u 1· … · u k = v 1 · … · v m . We show that for a large class of noetherian domains satisfying some natural finiteness conditions, the sets Vk (H){\cal V}_k (H) are almost arithmetical progressions. Suppose that H is a Krull monoid with finite cyclic class group G such that every class contains a prime (this includes the multiplicative monoids of rings of integers of algebraic number fields). We show that, for every k ? \Bbb Nk \in {\Bbb N}, max V2k+1 (H) = k |G|+ 1{\cal V}_{2k+1} (H) = k \vert G\vert + 1 which settles Problem 38 in [4].      10. Representations of nilpotent Lie algebras      Rolf Farnsteiner 《Archiv der Mathematik》1999,72(1):28-39 Let (L,[p]) a finite dimensional nilpotent restricted Lie algebra of characteristic p 3 3, c ? L*p \geq 3, \chi \in L^* a linear form. In this paper we study the representation theory of the reduced universal enveloping algebra u(L,c)u(L,\chi ). It is shown that u(L,c)u(L,\chi ) does not admit blocks of tame representation type. As an application, we prove that the nonregular AR-components of u(L,c)u(L,\chi ) are of types \Bbb Z [A￥ ]\Bbb Z [A_\infty ] or \Bbb Z [An]/(t)\Bbb Z [A_n]/(\tau ).      11. Refined asymptotics for the blowup of ut — δu = up      Stathis Filippas  Robert V. Kohn 《纯数学与应用数学通讯》1992,45(7):821-869 This work is concerned with positive, blowing-up solutions of the semilinear heat equation ut — δu = up in Rn. Our main contribution is a sort of center manifold analysis for the equation in similarity variables, leading to refined asymptotics for u in a backward space-time parabola near any blowup point. We also explore a connection between the asymptotics of u and the local geometry of the blowup set.      12. Banach Spaces of Solutions of the Helmholtz Equation in the Plane      Josefina Alvarez  Magali Folch-Gabayet  Salvador Pérez-Esteva 《Journal of Fourier Analysis and Applications》2001,7(1):49-62 The purpose of this article is to study the Hilbert space W2\mathcal{ W}^2 consisting of all solutions of the Helmholtz equation Du+u=0\Delta u+u=0 in \BbbR2\Bbb{R}^2 that are the image under the Fourier transform of L2L^2 densities in the unit circle. We characterize this space as a close subspace of the Hilbert space H2\mathcal{ H}^2 of all functions belonging to L2( | x | -3dx) L^2( | x | ^{-3}dx) jointly with their angular and radial derivatives, in the complement of the unit disk in \BbbR2\Bbb{R}^2. We calculate the reproducing kernel of W2\mathcal{ W}^2 and study its reproducing properties in the corresponding spaces Hp\mathcal{H}^p, for $p>1$p>1.      13. Thermal avalanche for blowup solutions of semilinear heat equations      Fernando Quirs  Julio D. Rossi  Juan Luis Vzquez 《纯数学与应用数学通讯》2004,57(1):59-98 We consider the semilinear heat equation ut = Δu + up both in ?N and in a bounded domain with homogeneous Dirichlet boundary conditions, with 1 < p < ps where ps is the Sobolev exponent. This problem has solutions with finite‐time blowup; that is, for large enough initial data there exists T < ∞ such that u is a classical solution for 0 < t < T, while it becomes unbounded as t ↗ T. In order to understand the situation for t > T, we consider a natural approximation by reaction problems with global solution and pass to the limit. As is well‐known, the limit solution undergoes complete blowup: after it blows up at t = T, the continuation is identically infinite for all t > T. We perform here a deeper analysis of how complete blowup occurs. Actually, the singularity set of a solution that blows up as t ↗ T propagates instantaneously at time t = T to cover the whole space, producing a catastrophic discontinuity between t = T? and t = T+. This is called the “avalanche.” We describe its formation as a layer appearing in the limit of the natural approximate problems. After a suitable scaling, the initial structure of the layer is given by the solution of a limit problem, described by a simple ordinary differential equation. As t proceeds past T, the solutions of the approximate problems have a traveling wave behavior with a speed that we compute. The situation in the inner region depends on the type of approximation: a conical pattern is formed with time when we approximate the power up by a flat truncation at level n, while for tangent truncations we get an exponential increase in time and a diffusive spatial pattern. © 2003 Wiley Periodicals, Inc.      14. Concentration at curves for a singularly perturbed Neumann problem in three-dimensional domains      A. Malchiodi 《Geometric And Functional Analysis》2005,15(6):1162-1222 We prove new concentration phenomena for the equation −ɛ2 Δu + u = up in a smooth bounded domain and with Neumann boundary conditions. Here p > 1 and ɛ > 0 is small. We show that concentration of solutions occurs at some geodesics of ∂Ω when ɛ → 0. Received: September 2004 Accepted: March 2005      15. Some blow-up results for a generalized Ginzburg-Landau equation      M. Aassila 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2002,53(1):1-11 We investigate the blow-up of the solution to a complex Ginzburg-Landau like equation in u coupled with a Poisson equation in f\phi defined on the whole space \Bbb Rn, n = 1{\Bbb R}^n, n = 1 or 2.      16. Finite Time Blow-up for a Non-linear Parabolic Equation with a Gradient Term and Applications      Philippe Souplet 《Mathematical Methods in the Applied Sciences》1996,19(16):1317-1333 We give new finite time blow-up results for the non-linear parabolic equations ut−Δu = up and ut−Δu+μ∣∇u∣q = up. We first establish an a priori bound in Lp+1 for the positive non-decreasing global solutions. As a consequence, we prove in particular that for the second equation on ℝN, with q = 2p/(p+1) and small μ>0, blow-up can occur for any N≥1, p>1, (N−2)p<N+2 and without energy restriction on the initial data. Incidentally, we present a simple model in population dynamics involving this equation.      17. Large solutions of elliptic equations with a weakly superlinear nonlinearity      Florica Corina Cîrstea  Yihong Du 《Journal d'Analyse Mathématique》2007,103(1):261-277 This paper studies the asymptotic behavior near the boundary for large solutions of the semilinear equation Δu + au = b(x)f(u) in a smooth bounded domain Ω of ℝN with N ≥ 2, where a is a real parameter and b is a nonnegative smooth function on . We assume that f(u) behaves like u(ln u)α as u → ∞, for some α > 2. It turns out that this case is more difficult to handle than those where f(u) grows like u p (p > 1) or faster at infinity. Under suitable conditions on the weight function b(x), which may vanish on ∂Ω, we obtain the first order expansion of the large solutions near the boundary. We also obtain some uniqueness results. Research of both authors supported by the Australian Research Council.      18. Euler‐lagrange equation and regularity for flat minimizers of the Willmore functional      Peter Hornung 《纯数学与应用数学通讯》2011,64(3):367-441 Let \input amssym $S\subset{\Bbb R}^2$ be a bounded domain with boundary of class C∞, and let gij = δij denote the flat metric on \input amssym ${\Bbb R}^2$ . Let u be a minimizer of the Willmore functional within a subclass (defined by prescribing boundary conditions on parts of ∂S) of all W2,2 isometric immersions of the Riemannian manifold (S, g) into \input amssym ${\Bbb R}^3$ . In this article we derive the Euler‐Lagrange equation and study the regularity properties for such u. Our main regularity result is that minimizers u are C3 away from a certain singular set Σ and C∞ away from a larger singular set Σ ∪ Σ0. We obtain a geometric characterization of these singular sets, and we derive the scaling of u and its derivatives near Σ0. Our main motivation to study this problem comes from nonlinear elasticity: On isometric immersions, the Willmore functional agrees with Kirchhoff's energy functional for thin elastic plates. © 2010 Wiley Periodicals, Inc.      19. On critical Fujita exponents for heat equations with nonlinear flux conditions on the boundary   总被引：1，自引：0，他引：1   Victor A. Galaktionov  Howard A. Levine 《Israel Journal of Mathematics》1996,94(1):125-146 We consider nonnegative solutions of initial-boundary value problems for parabolic equationsu t=uxx, ut=(um)xxand (m>1) forx>0,t>0 with nonlinear boundary conditions−u x=up,−(u m)x=upand forx=0,t>0, wherep>0. The initial function is assumed to be bounded, smooth and to have, in the latter two cases, compact support. We prove that for each problem there exist positive critical valuesp 0,pc(withp 0<pc)such that forp∃(0,p 0],all solutions are global while forp∃(p0,pc] any solutionu≢0 blows up in a finite time and forp>p csmall data solutions exist globally in time while large data solutions are nonglobal. We havep c=2,p c=m+1 andp c=2m for each problem, whilep 0=1,p 0=1/2(m+1) andp 0=2m/(m+1) respectively. This work was done during visits of the first author to Iowa State University and the Institute for Mathematics and its Applications at the University of Minnesota. The second author was supported in part by NSF Grant DMS-9102210.      20. A Modification of the Blowup Technique for Variational Integrals with Subquadratic Growth      Martin Fuchs  Jürgen Reuling 《Journal of Mathematical Analysis and Applications》1997,210(2):484 We present a version of the blowup technique which applies to local minimizersu: Ω → N, Ω open in n, of strictly convex variational integralsJ(u) ∫Ω f(u) dx, wherefis of p-growth for some 1 < p < 2. This provides an alternative approach towards the partial regularity theorem of Anzellotti and Giaquinta. For two-dimensional problems we obtain everywhereC1, α-regularity.

On products of k atoms

Let H be an atomic monoid. For k ? \Bbb Nk \in {\Bbb N} let V_k (H){\cal V}_k (H) denote the set of all m ? \Bbb Nm \in {\Bbb N} with the following property: There exist atoms (irreducible elements) u ₁, …, u _k , v ₁, …, v _m ∈ H with u ₁· … · u _k = v ₁ · … · v _m . We show that for a large class of noetherian domains satisfying some natural finiteness conditions, the sets V_k (H){\cal V}_k (H) are almost arithmetical progressions. Suppose that H is a Krull monoid with finite cyclic class group G such that every class contains a prime (this includes the multiplicative monoids of rings of integers of algebraic number fields). We show that, for every k ? \Bbb Nk \in {\Bbb N}, max V_2k+1 (H) = k |G|+ 1{\cal V}_{2k+1} (H) = k \vert G\vert + 1 which settles Problem 38 in [4].     

Representations of nilpotent Lie algebras

Let (L,[p]) a finite dimensional nilpotent restricted Lie algebra of characteristic p 3 3, c ? L^*p \geq 3, \chi \in L^* a linear form. In this paper we study the representation theory of the reduced universal enveloping algebra u(L,c)u(L,\chi ). It is shown that u(L,c)u(L,\chi ) does not admit blocks of tame representation type. As an application, we prove that the nonregular AR-components of u(L,c)u(L,\chi ) are of types \Bbb Z [A_￥ ]\Bbb Z [A_\infty ] or \Bbb Z [A_n]/(t)\Bbb Z [A_n]/(\tau ).     

Refined asymptotics for the blowup of ut — δu = up

This work is concerned with positive, blowing-up solutions of the semilinear heat equation u_t — δu = u^p in Rⁿ. Our main contribution is a sort of center manifold analysis for the equation in similarity variables, leading to refined asymptotics for u in a backward space-time parabola near any blowup point. We also explore a connection between the asymptotics of u and the local geometry of the blowup set.     

Banach Spaces of Solutions of the Helmholtz Equation in the Plane

The purpose of this article is to study the Hilbert space W²\mathcal{ W}^2 consisting of all solutions of the Helmholtz equation Du+u=0\Delta u+u=0 in \BbbR²\Bbb{R}^2 that are the image under the Fourier transform of L²L^2 densities in the unit circle. We characterize this space as a close subspace of the Hilbert space H²\mathcal{ H}^2 of all functions belonging to L²( | x | ^-3dx) L^2( | x | ^{-3}dx) jointly with their angular and radial derivatives, in the complement of the unit disk in \BbbR²\Bbb{R}^2. We calculate the reproducing kernel of W²\mathcal{ W}^2 and study its reproducing properties in the corresponding spaces H^p\mathcal{H}^p, for $p>1$p>1.     

Thermal avalanche for blowup solutions of semilinear heat equations

We consider the semilinear heat equation u_t = Δu + u^p both in ?^N and in a bounded domain with homogeneous Dirichlet boundary conditions, with 1 < p < p_s where p_s is the Sobolev exponent. This problem has solutions with finite‐time blowup; that is, for large enough initial data there exists T < ∞ such that u is a classical solution for 0 < t < T, while it becomes unbounded as t ↗ T. In order to understand the situation for t > T, we consider a natural approximation by reaction problems with global solution and pass to the limit. As is well‐known, the limit solution undergoes complete blowup: after it blows up at t = T, the continuation is identically infinite for all t > T. We perform here a deeper analysis of how complete blowup occurs. Actually, the singularity set of a solution that blows up as t ↗ T propagates instantaneously at time t = T to cover the whole space, producing a catastrophic discontinuity between t = T^? and t = T⁺. This is called the “avalanche.” We describe its formation as a layer appearing in the limit of the natural approximate problems. After a suitable scaling, the initial structure of the layer is given by the solution of a limit problem, described by a simple ordinary differential equation. As t proceeds past T, the solutions of the approximate problems have a traveling wave behavior with a speed that we compute. The situation in the inner region depends on the type of approximation: a conical pattern is formed with time when we approximate the power u^p by a flat truncation at level n, while for tangent truncations we get an exponential increase in time and a diffusive spatial pattern. © 2003 Wiley Periodicals, Inc.     

Concentration at curves for a singularly perturbed Neumann problem in three-dimensional domains

We prove new concentration phenomena for the equation −ɛ² Δu + u = u^p in a smooth bounded domain and with Neumann boundary conditions. Here p > 1 and ɛ > 0 is small. We show that concentration of solutions occurs at some geodesics of ∂Ω when ɛ → 0. Received: September 2004 Accepted: March 2005     

Some blow-up results for a generalized Ginzburg-Landau equation

We investigate the blow-up of the solution to a complex Ginzburg-Landau like equation in u coupled with a Poisson equation in f\phi defined on the whole space \Bbb Rⁿ, n = 1{\Bbb R}^n, n = 1 or 2.     

Finite Time Blow-up for a Non-linear Parabolic Equation with a Gradient Term and Applications

We give new finite time blow-up results for the non-linear parabolic equations u_t−Δu = u^p and u_t−Δu+μ∣∇u∣^q = u^p. We first establish an a priori bound in L^p+1 for the positive non-decreasing global solutions. As a consequence, we prove in particular that for the second equation on ℝ^N, with q = 2p/(p+1) and small μ>0, blow-up can occur for any N≥1, p>1, (N−2)p<N+2 and without energy restriction on the initial data. Incidentally, we present a simple model in population dynamics involving this equation.     

Large solutions of elliptic equations with a weakly superlinear nonlinearity

This paper studies the asymptotic behavior near the boundary for large solutions of the semilinear equation Δu + au = b(x)f(u) in a smooth bounded domain Ω of ℝ^N with N ≥ 2, where a is a real parameter and b is a nonnegative smooth function on . We assume that f(u) behaves like u(ln u)^α as u → ∞, for some α > 2. It turns out that this case is more difficult to handle than those where f(u) grows like u ^p (p > 1) or faster at infinity. Under suitable conditions on the weight function b(x), which may vanish on ∂Ω, we obtain the first order expansion of the large solutions near the boundary. We also obtain some uniqueness results. Research of both authors supported by the Australian Research Council.     

Euler‐lagrange equation and regularity for flat minimizers of the Willmore functional

Let \input amssym $S\subset{\Bbb R}^2$ be a bounded domain with boundary of class C^∞, and let g_ij = δ_ij denote the flat metric on \input amssym ${\Bbb R}^2$ . Let u be a minimizer of the Willmore functional within a subclass (defined by prescribing boundary conditions on parts of ∂S) of all W^2,2 isometric immersions of the Riemannian manifold (S, g) into \input amssym ${\Bbb R}^3$ . In this article we derive the Euler‐Lagrange equation and study the regularity properties for such u. Our main regularity result is that minimizers u are C³ away from a certain singular set Σ and C^∞ away from a larger singular set Σ ∪ Σ₀. We obtain a geometric characterization of these singular sets, and we derive the scaling of u and its derivatives near Σ₀. Our main motivation to study this problem comes from nonlinear elasticity: On isometric immersions, the Willmore functional agrees with Kirchhoff's energy functional for thin elastic plates. © 2010 Wiley Periodicals, Inc.     

On critical Fujita exponents for heat equations with nonlinear flux conditions on the boundary

We consider nonnegative solutions of initial-boundary value problems for parabolic equationsu _t=u_xx, u_t=(u^m)_xxand (m>1) forx>0,t>0 with nonlinear boundary conditions−u _x=u^p,−(u ^m)_x=u^pand forx=0,t>0, wherep>0. The initial function is assumed to be bounded, smooth and to have, in the latter two cases, compact support. We prove that for each problem there exist positive critical valuesp ₀,p_c(withp ₀<p_c)such that forp∃(0,p ₀],all solutions are global while forp∃(p₀,p_c] any solutionu≢0 blows up in a finite time and forp>p _csmall data solutions exist globally in time while large data solutions are nonglobal. We havep _c=2,p _c=m+1 andp _c=2m for each problem, whilep ₀=1,p ₀=1/2(m+1) andp ₀=2m/(m+1) respectively. This work was done during visits of the first author to Iowa State University and the Institute for Mathematics and its Applications at the University of Minnesota. The second author was supported in part by NSF Grant DMS-9102210.     

A Modification of the Blowup Technique for Variational Integrals with Subquadratic Growth

We present a version of the blowup technique which applies to local minimizersu: Ω → ^N, Ω open in ⁿ, of strictly convex variational integralsJ(u) ∫_Ω f(u) dx, wherefis of p-growth for some 1 < p < 2. This provides an alternative approach towards the partial regularity theorem of Anzellotti and Giaquinta. For two-dimensional problems we obtain everywhereC^{1, α}-regularity.