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  • 1.
    Ahmadzadeh, Farzaneh
    et al.
    Luleå University of Technology, Department of Civil, Environmental and Natural Resources Engineering, Operation, Maintenance and Acoustics.
    Lundberg, Jan
    Luleå University of Technology, Department of Civil, Environmental and Natural Resources Engineering, Operation, Maintenance and Acoustics.
    Strömberg, Thomas
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Multivariate process parameter change identification by neural network2013In: The International Journal of Advanced Manufacturing Technology, ISSN 0268-3768, E-ISSN 1433-3015, Vol. 69, no 9-12, p. 2261-2268Article in journal (Refereed)
    Abstract [en]

    Whenever there is an out-of-control signal in process parameter control charts, maintenance engineers try to diagnose the cause near the time of the signal which is not always lead to prompt identification of the source(s) of the out-of-control condition and this in some cases yields to extremely high monetary loses for manufacture owner. This paper applies multivariate exponentially weighted moving average (MEWMA) control charts and neural networks to make the signal identification more effective. The simulation of this procedure shows that this new control chart can be very effective in detecting the actual change point for all process dimension and all shift magnitudes considered. This methodology can be used in manufacturing and process industries to predict change points and expedite the search for failure causing parameters, resulting in improved quality at reduced overall cost. This research shows development of MEWMA by usage of neural network for identifying the step change point and the variable responsible for the change in the process mean vector.

  • 2.
    Berglund, Tomas
    et al.
    Luleå University of Technology, Department of Computer Science, Electrical and Space Engineering, Embedded Internet Systems Lab.
    Strömberg, Thomas
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Jonsson, Håkan
    Luleå University of Technology, Department of Computer Science, Electrical and Space Engineering, Computer Science.
    Söderkvist, Inge
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Epi-convergence of minimum curvature variation B-splines2003Report (Other academic)
    Abstract [en]

    We study the curvature variation functional, i.e., the integral over the square of arc-length derivative of curvature, along a planar curve. With no other constraints than prescribed position, slope angle, and curvature at the endpoints of the curve, the minimizer of this functional is known as a cubic spiral. It remains a challenge to effectively compute minimizers or approximations to minimizers of this functional subject to additional constraints such as, for example, for the curve to avoid obstacles such as other curves. In this paper, we consider the set of smooth curves that can be written as graphs of three times continuously differentiable functions on an interval, and, in particular, we consider approximations using quartic uniform B- spline functions. We show that if quartic uniform B-spline minimizers of the curvature variation functional converge to a curve, as the number of B-spline basis functions tends to infinity, then this curve is in fact a minimizer of the curvature variation functional. In order to illustrate this result, we present an example of sequences of B-spline minimizers that converge to a cubic spiral.

  • 3.
    Choquard, Philippe
    et al.
    EPFL, Lausanne.
    Strömberg, Thomas
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    A one-dimensional inviscid and compressible fluid in a harmonic potential well2007In: Acta Applicandae Mathematicae - An International Survey Journal on Applying Mathematics and Mathematical Applications, ISSN 0167-8019, E-ISSN 1572-9036, Vol. 99, no 2, p. 161-183Article in journal (Refereed)
    Abstract [en]

    A Hamiltonian model is analyzed for a one-dimensional inviscid compressible fluid. The space-time evolution of the fluid is governed by the following system of the Hamilton-Jacobi and the continuity equations: S-t + 1/2(S-x(2) + omega(2)chi(2)) =0, S(x, 0) = S-0(x); rho(t) + (rho S-x)(x) =0, rho(x, 0) =rho(0)(x).Here S and rho designate the velocity potencial and the mass density, respectively. Unless S-0 is convex, shocks form and the velocity S (x) becomes discontinuous in {0 < omega t < pi/2}. It is demonstrated that there nevertheless exists a unique viscosity-measure solution (S,rho) when S-0 is globally Lipschitz continuous and locally semi-concave while rho(0) is a finite Borel measure. The structure of the velocity and the density is exhibited. For initial data correlated in a certain sense, a class of classical solutions (S,rho) is given. Negative time is also considered, and illustrating examples are given.

  • 4.
    Johnsson, Helena
    et al.
    Luleå University of Technology, Department of Civil, Environmental and Natural Resources Engineering, Structural and Construction Engineering.
    Alerby, Eva
    Luleå University of Technology, Department of Arts, Communication and Education, Education, Language, and Teaching.
    Hyyppä, Kalevi
    Luleå University of Technology, Department of Computer Science, Electrical and Space Engineering, Embedded Internet Systems Lab.
    Jonsson, Håkan
    Luleå University of Technology, Department of Computer Science, Electrical and Space Engineering, Computer Science.
    Karlberg, Magnus
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Product and Production Development.
    Stenberg, Magnus
    Luleå University of Technology, Department of Business Administration, Technology and Social Sciences, Human Work Science.
    Strömberg, Thomas
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    En pedagogisk idé för ingenjörsutbildningarna vid Luleå tekniska universitet2009Conference paper (Refereed)
    Abstract [sv]

    Utmärkande för Luleå tekniska universitet (LTU) har länge varit närheten mellan studenterna och lärarna vilket har manifesterats genom klassrumsundervisning, lärartillgänglighet, mindre studiegrupper, koncentrerade campus etc. Undervisningsformerna har dock ändrats till att idag omfatta dels klassiska föreläsningar i stora studentgrupper, men också projektarbete i team där totalt sett 1/3 av all undervisning vid Luleå tekniska universitet sker i projektform. En fråga har ställts: Har Luleå tekniska universitet idag en gemensam pedagogisk idé för ingenjörsutbildningarna? I så fall är ytterligare en fråga hur denna pedagogiska idé tar sig uttryck i organisationen?Intervjuer med 40 aktiva lärare vid Luleå tekniska universitet har genomförts av projektgruppen (tillika författarna) varpå svaren har grupperats och analyserats i fjorton olika teman. Från dessa teman har sedan fyra hörnstenar i en definition av en pedagogisk idé aggregerats. Formuleringen av den pedagogiska idén lyder: Ett aktivt lärande för yrkeslivet – i branschnära projekt och med god vetenskaplig grund tränas förmågan att arbeta som ingenjör genom coachning från lärare i ett nära och öppet klimat. Idén har antagits av den tekniska fakultetsnämnden vid Luleå tekniska universitet och verifierats på institutionerna. Exempel på implementering och hur den pedagogiska idén aktivt verkar presenteras i artikeln.

  • 5. Kjellmert, Bo
    et al.
    Strömberg, Thomas
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Time-dependent electromagnetic waves in a cavity2009In: Applications of Mathematics, ISSN 0862-7940, E-ISSN 1572-9109, Vol. 54, no 1, p. 17-45Article in journal (Refereed)
    Abstract [en]

    The electromagnetic initial-boundary value problem for a cavity enclosed by perfectly conducting walls is considered. The cavity medium is defined by its permittivity and permeability which vary continuously in space. The electromagnetic field comes from a source in the cavity. The field is described by a magnetic vector potential A satisfying a wave equation with initial-boundary conditions. This description through A is rigorously shown to give a unique solution of the problem and is the starting point for numerical computations. A Chebyshev collocation solver has been implemented for a cubic cavity, and it has been compared to a standard finite element solver. The results obtained are consistent while the collocation solver performs substantially faster. Some time histories and spectra are computed.

  • 6.
    Maad, Sara
    et al.
    University of Virginia.
    Martin, Clyde
    Texas Tech University.
    Strömberg, Thomas
    Byström, Johan
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Surface fitting with boundary data2004In: 2004 43rd IEEE Conference on Decision and Control: Nassau, Bahamas, 14 - 17 December 2004, Piscataway, NJ: IEEE Communications Society, 2004, Vol. Vol. 4, p. 3649-3653Conference paper (Refereed)
    Abstract [en]

    The problem of fitting surfaces to data is a well studied problem in statistics. However when there is prior information the theory is not developed. It often happens in toxicology and in medicine that the effect of a single drug is well understood.. However if a pair of drugs is delivered in tandem or if two toxins are interacting the effect is not understood. In this paper we attack the problem of fitting a surface to a data set contained in a square when two of the boundaries are known. Our approach generalizes the concept of smoothing splines.

  • 7.
    Persson, Lars-Erik
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Strömberg, Thomas
    Green's method applied to the plate equation in mechanics1993In: Annales Societatis Mathematicae Polonae. Series 1: Commentationes Mathematicae - Prace Matematyczne, ISSN 0373-8299, Vol. 33, p. 119-133Article in journal (Refereed)
    Abstract [en]

    This paper demonstrates existence and uniqueness of Green functions for the general fourth-order linear elliptic operator modeling the deflection of an anisotropic elastic plate with the linear boundary conditions that model the most common edge conditions. The existence theory is presented in the context of Sobolev spaces. Included is the derivation of the boundary value problem from the principles of elasticity and of the Green function for a homogeneous rectangular plate. The paper concludes with a brief discussion of qualitative properties of the Green functions. The techniques and results are classic, and the exposition is accessible to a broad audience. This paper could serve as an admirable introduction to the boundary value problems of anisotropic plates.

  • 8.
    Strömberg, Thomas
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    A counterexample to uniqueness of generalized characteristics in Hamilton-Jacobi theory2011In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 74, no 7, p. 2758-2762Article in journal (Refereed)
    Abstract [en]

    The notion of generalized characteristics plays a pivotal role in the study of propagation of singularities for Hamilton{Jacobi equations. This note gives an example of nonuniqueness of forward generalized characteristics emanating from a given point.

  • 9.
    Strömberg, Thomas
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    A new proof of indefinite propagation of singularities for a Hamilton-Jacobi equation2016In: Journal of evolution equations (Printed ed.), ISSN 1424-3199, E-ISSN 1424-3202, Vol. 16, no 4, p. 895-903Article in journal (Refereed)
    Abstract [en]

    We study propagation of singularities for the Hamilton–Jacobi equation S t +H(∇S)=0,(t,x)∈(0,T)×R n , St+H(∇S)=0,(t,x)∈(0,T)×Rn,where H(p)=12 ⟨p,Ap⟩ H(p)=12⟨p,Ap⟩ is a positive definite quadratic form. Each viscosity solution S S is semiconcave, and it is known that its singularities move along generalized characteristics. We give a new proof of the recent result by Cannarsa et al. (Discrete Contin Dyn Syst 35:4225–4239, 2015), namely that the singularities propagate along generalized characteristics indefinitely forward in time. 

  • 10.
    Strömberg, Thomas
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    A note on the differentiability of conjugate functions2009In: Archiv der Mathematik, ISSN 0003-889X, E-ISSN 1420-8938, Vol. 93, no 5, p. 481-485Article in journal (Refereed)
    Abstract [en]

    For a proper, lower semicontinuous and convex function f with Legendre-Fenchel conjugate f *, it is well-known that differentiability properties of f * are equivalent to strict convexity properties of f. In this note a result of this kind is obtained without a convexity assumption on f.

  • 11.
    Strömberg, Thomas
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    A study of the operation of infimal convolution1994Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    This thesis consists of five papers (A-E), which examine the operation of infimal convolution and discuss its close connections to unilateral analysis, convex analysis, inequalities, approximation, and optimization. In particular, we attempt to provide a detailed investigation for both the convex and the non-convex case, including several examples. Paper (A) is both a survey of and a self-contained introduction to the operation of infimal convolution. In particular, we discuss the infimal value and minimizers of an infimal convolute, infimal convolution on subadditive functions, sufficient conditions for semicontinuity or continuity of an infimal convolute, "exactness," regularizing effects, continuity of the operation of infimal convolution, and approximation methods based on infimal convolution. A Young-type inequality, closely connected to the operation of infimal convolution, is studied in paper (B). The main results obtained are an equivalence theorem and a representation formula. In paper (C) we consider coercive, convex, proper, and lower sernicontinuous functions on a reflexive Banach space. For the infimal convolution of such functions we establish, in particular, different formulae. Moreover, we demonstrate the possibility of using the formulae obtained for solving special types of Hamilton-Jacobi equations. Furthermore, the operation of infimal convolution is interpreted from a physical viewpoint. Paper (D) presents properties of infimal convolution of functions that are uniformly continuous on bounded sets. In particular, we present regularization procedures by means of infimal convolution. The role of growth conditions on the functions under consideration is essential. Finally, in paper (E) we study semicontinuity, continuity, and differentiability of the infimal convolute of two convex functions. Moreover, under certain geometric conditions, the classical Moreau-Yosida approximation process is, roughly speaking, extended to the non-convex case.

  • 12.
    Strömberg, Thomas
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    A system of the Hamilton-Jacobi and the continuity equations in the vanishing viscosity limit2011In: Communications on Pure and Applied Analysis, ISSN 1534-0392, E-ISSN 1553-5258, Vol. 10, no 2, p. 479-506Article in journal (Refereed)
    Abstract [en]

    We study the following system of the viscous Hamilton-Jacobi and the continuity equations in the limit as epsilon down arrow 0: S-t(epsilon) + 1/2 vertical bar DS epsilon vertical bar(2) + V(x) - epsilon Delta S-epsilon = 0 in Q(T), S-epsilon(0, x) = S-0(x) in R-n; rho(epsilon)(t) + div(rho(epsilon) DS epsilon) = 0 in Q(T), rho(epsilon)(0, x) = rho(0)(x) in R-n. Here Q(T) = (0, T] x R-n. The potential V and the initial function S-0 are allowed to grow quadratically while rho(0) is a Borel measure. The paper justifies and describes the vanishing viscosity transition to the corresponding inviscid system. The notion of weak solution employed for the inviscid system is that of a viscosity-measure solution (S, rho).

  • 13.
    Strömberg, Thomas
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    An operation connected to a Young-type inequality1992In: Mathematische Nachrichten, ISSN 0025-584X, E-ISSN 1522-2616, Vol. 159, p. 227-243Article in journal (Refereed)
  • 14.
    Strömberg, Thomas
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Duality between Frechet differentiability and strong convexity2011In: Positivity (Dordrecht), ISSN 1385-1292, E-ISSN 1572-9281, Vol. 15, no 3, p. 527-536Article in journal (Refereed)
    Abstract [en]

    This paper revisits the duality between differentiability and strict or strong convexity under the Legendre-Fenchel transform {Mathematical expression}. Functions f defined on a Banach space X are considered. For a lower semicontinuous but not necessarily convex function f we relate essential Fréchet differentiability of the conjugate function f* to essential strong convexity of f

  • 15.
    Strömberg, Thomas
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Exponentially growing solutions of parabolic Isaacs' equations2008In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 348, no 1, p. 337-345Article in journal (Refereed)
    Abstract [en]

    This paper contributes to the literature on unbounded viscosity solutions of fully nonlinear and possibly degenerate parabolic equations. Under natural assumptions, it is established that the initial-value problem for a degenerate parabolic Isaacs' equation set in (0, T] × R has a unique continuous viscosity solution with at most exponential growth at infinity.

  • 16.
    Strömberg, Thomas
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Hamilton-Jacobi equations having only action functions as solutions2004In: Archiv der Mathematik, ISSN 0003-889X, E-ISSN 1420-8938, Vol. 83, no 5, p. 437-449Article in journal (Refereed)
    Abstract [en]

    Let L(x, v) be a Lagrangian which is convex and superlinear in the velocity variable v, and let H(x, p) be the associated Hamiltonian. Conditions are obtained under which every viscosity solution ....

  • 17.
    Strömberg, Thomas
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Moreau envelope function2000In: Encyclopaedia of mathematics: an updated and annotated translation of the Soviet ’Mathematical encyclopaedia’, Dordrecht: Encyclopedia of Global Archaeology/Springer Verlag, 2000, p. 346-347Chapter in book (Other academic)
  • 18.
    Strömberg, Thomas
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    On a degenerate Hamilton-Jacobi equation and a related topic2002In: International journal of pure and applied mathematics, ISSN 1311-8080, E-ISSN 1314-3395, Vol. 3, no 3, p. 325-342Article in journal (Refereed)
  • 19.
    Strömberg, Thomas
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    On a viscous Hamilton-Jacobi equation with an unbounded potential term2010In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 73, no 6, p. 1802-1811Article in journal (Refereed)
    Abstract [en]

    We present comparison, uniqueness and existence results for unbounded solutions of a viscous Hamilton-Jacobi or eikonal equation. The equation includes an unbounded potential term V(x) subject to a quadratic upper bound. The results are obtained through a tailor-made change of variables in combination with the Hopf-Cole transformation. An integral representation formula for the solution of the Cauchy problem is derived in the case where V(x)=ω2x2/2

  • 20.
    Strömberg, Thomas
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    On infimal convolution and a related operation connected to a Young-type inequality with applications1992Licentiate thesis, comprehensive summary (Other academic)
  • 21. Strömberg, Thomas
    On regularization in Banach spaces1996In: Arkiv för matematik, ISSN 0004-2080, E-ISSN 1871-2487, Vol. 34, no 2, p. 383-406Article in journal (Refereed)
  • 22.
    Strömberg, Thomas
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    On shock generation for Hamilton-Jacobi equations2011In: Indagationes mathematicae, ISSN 0019-3577, E-ISSN 1872-6100, Vol. 20, 2009, no 4, p. 619-629Article in journal (Refereed)
    Abstract [en]

    The subject of this paper is the generation of singularities of solutions of Hamilton-Jacobi equations set in (0, ∞) × fordataofclass C∞. Shockwaves originate from conjugate points. To show sharpness of a known Hausdorff estimate, an example is given in which the set of conjugate, regular points includes uncountably many affine subspaces of dimension n − 1.

  • 23.
    Strömberg, Thomas
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    On the epigraphical sum of functions uniformly continuous on bounded sets1992In: Séminaire d'Analyse Convexe, Vol. 22, no 19Article in journal (Refereed)
  • 24.
    Strömberg, Thomas
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    On the Hamilton-Jacobi equation for a harmonic oscillator2010In: Results in Mathematics, ISSN 1422-6383, Vol. 57, no 3-4, p. 195-204Article in journal (Refereed)
    Abstract [en]

    The topic of this note is the classical Hamilton-Jacobi equation {Mathematical expression}In complete generality, a description of the superdifferential {Mathematical expression} of the viscosity solution of the initial-value problem for this equation is furnished in terms of a convex hull construction and rotations. For background, the basic existence and uniqueness properties of the viscosity solution S are recalled.

  • 25.
    Strömberg, Thomas
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    On viscosity solutions of irregular Hamilton-Jacobi equations2003In: Archiv der Mathematik, ISSN 0003-889X, E-ISSN 1420-8938, Vol. 81, no 6, p. 678-688Article in journal (Refereed)
    Abstract [en]

    In this paper, the author proves the existence and uniqueness of continuous viscosity solutions for some Hamilton-Jacobi equations with possibly discontinuous Hamiltonians. As a corollary, he gets the existence and uniqueness of continuous viscosity solutions for some transport equations with discontinuous coefficients.

  • 26.
    Strömberg, Thomas
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    On viscosity solutions of the Hamilton-Jacobi equation1999In: Hokkaido Mathematical Journal, ISSN 0385-4035, Vol. 28, no 3, p. 475-506Article in journal (Refereed)
  • 27.
    Strömberg, Thomas
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Propagation of singularities along broken characteristics2013In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 85, p. 93-109Article in journal (Refereed)
    Abstract [en]

    This paper contributes to the analysis of propagation of singularities for semiconcave solutions of Hamilton–Jacobi equations. Under certain conditions, we establish the existence and uniqueness of certain broken characteristics termed strong characteristics. If u is a viscosity solution of a Hamilton–Jacobi equation, then strong characteristics carry the singularities of u.

  • 28.
    Strömberg, Thomas
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Representation formulae for infimal convolution with applications1994In: Analysis, algebra, and computers in mathematical research: proceedings of the twenty-first Nordic Congress of Mathematicians / [ed] Mats Gyllenberg; Lars Erik Persson, New York: Marcel Dekker Incorporated , 1994, p. 319-334Conference paper (Refereed)
  • 29.
    Strömberg, Thomas
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Semiconcavity estimates for viscous Hamilton-Jacobi equations2010In: Archiv der Mathematik, ISSN 0003-889X, E-ISSN 1420-8938, Vol. 94, no 6, p. 579-589Article in journal (Refereed)
    Abstract [en]

    We present sharp Hessian estimates of the form {Mathematical expression} for the solution of the viscous Hamilton-Jacobi equation {Mathematical expression}The smallest possible positive function g(t) is explicitly given in terms of the semiconvexity and semiconcavity parameters of V and S0, respectively. The optimal g does not depend on the viscosity parameter {Mathematical expression} . The potential V and the initial function S0 are allowed to grow quadratically

  • 30.
    Strömberg, Thomas
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    The Hopf-Lax formula gives the unique viscosity solution2002In: Differential and Integral Equations, ISSN 0893-4983, Vol. 15, no 1, p. 47-52Article in journal (Refereed)
  • 31. Strömberg, Thomas
    The operation of infimal convolution1996Report (Other academic)
    Abstract [en]

    In this well-written paper on infimal convolution the author's purpose is "to provide a survey of the subject as well as to complement or sharpen existing results". The author has chosen a self-contained presentation, where, generally, proofs are given for his original results (spread in several articles). The paper is organized in five sections: 1. Introduction and preliminaries, 2. Elementary properties, 3. The convex case, 4. Continuity of the operation of infimal convolution, 5. Regularization. An important concern is the regularizing effect of the infimal convolution. So, under some additional hypotheses, $f\nabla g$ is (upper semi-) continuous, uniformly continuous (on bounded sets), Gateaux (Fr\'{e}chet) differentiable, with uniformly (or $p$-H\"{o}lder) continuous derivative if $g$ is so. The continuity properties of the operation of infimal convolution are studied with respect to the epi-convergence, the slice topology, the affine topology and the Attouch-Wets topology. The last section is dedicated to Moreau-Yosida and Lasry-Lions approximations as well as to a generalized Moreau-Yosida approximation. The results established in the previous sections are applied to these particular infimal convolutes, leading to several interesting results.

  • 32.
    Strömberg, Thomas
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Viscosity solutions of fully nonlinear PDEs2009In: AIHT : Analysis, Inequalities and Homogenization Theory: Midnight sun conference in honor of Lars-Erik Persson, 2009Conference paper (Other academic)
    Abstract [en]

    I discuss in talk the Cauchy problem for fully nonlinear and possibly degenerate parabolic equations of the form ut + F(t, x, u,Du,D2u) = 0 set in QT = (0, T] × Rn. The delicate uniqueness issue is the main topic. Since the PDE is set in an unbounded set, it is, generally speaking, necessary to impose restrictons on the growth on the solution to prove uniqueness. However, I mention uniqueness results without any restrictions, for (i) the inviscid Hamilon-Jacobi equation ut + H(t, x,Du) = 0 or for (ii) the viscous equation ut+ 1 2 |Du|2+V (x)-"u = 0. A result for the Isaacs equation under an exponential growth condition on the solution is also given.

  • 33.
    Strömberg, Thomas
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Well-posedness for the system of the Hamilton-Jacobi and the continuity equations2007In: Journal of evolution equations (Printed ed.), ISSN 1424-3199, E-ISSN 1424-3202, Vol. 7, no 4, p. 669-700Article in journal (Refereed)
    Abstract [en]

    Let H (t, x, p) be a Hamiltonian function that is convex in p. Let the associated Lagrangian satisfy the nonstandard minorization condition where m > 0, ω > 0, and C ≥ 0 are constants. Under some additional conditions, we prove that the associated value function is the unique viscosity solution of S t + H(t, x, ∇S) = 0 in , without any conditions at infinity on the solution. Here ωT < π/2. To the Hamilton-Jacobi equation corresponding to the classical action integrand in mechanics, we adjoin the continuity equation and establish the existence and uniqueness of a viscosity-measure solution (S, ρ) of ...This system arises in the WKB method. The measure solution is defined by means of the Filippov flow of ∇S.

  • 34.
    Strömberg, Thomas
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Ahmadzadeh, Farzaneh
    School of Innovation, Design and Engineering, Mälardalen University.
    Excess action and broken characteristics for Hamilton-Jacobi equations2014In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 110, p. 113-129Article in journal (Refereed)
    Abstract [en]

    We study propagation of singularities for Hamilton–Jacobi equations View the MathML sourceSt+H(t,x,∇S)=0,(t,x)∈(0,∞)×Rn,Turn MathJax onby means of the excess Lagrangian action and a related class of characteristics. In a sense, the excess action gauges how far a curve View the MathML sourceX(t) is from being action minimizing for a given viscosity solution S(t,x)S(t,x) of the Hamilton–Jacobi equation. Broken characteristics are defined as curves along which the excess action grows at the slowest pace possible. In particular, we demonstrate that broken characteristics carry the singularities of the viscosity solution.

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