## Abstract

We consider the inverse problem of determining the Winkler subgrade reaction coefficient of a slab foundation modelled as a thin elastic plate clamped at the boundary. The plate is loaded by a concentrated force and its transversal deflection is measured at the interior points. We prove a global Hölder stability estimate under (mild) regularity assumptions on the unknown coefficient.

## 1. Introduction

The soil–structure interaction is an important issue in structural building design. The determination of the contact actions exchanged between foundation and soil is commonly approached by using simplified models of interaction. Among these, the model introduced by Winkler [1] in the second half of the nineteenth century is one of the most popular in engineering and geotechnical applications [2]. In Winkler's model, the foundation rests on a bed of linearly elastic springs of stiffness *k*, *k*≥0, acting along the vertical direction only. The springs are independent of each other, that is, the deflection of every spring is not influenced by the other adjacent springs. The accuracy of this model of interaction depends strongly on the values assigned to the subgrade reaction coefficient *k*. Ranges of average values of *k* are available in the literature from extensive series of *in situ* experiments performed on various soil types [3], but these values are quite scattered and, in addition, they may vary significantly from point to point in the case of large foundations. Estimation of the coefficient *k* becomes even more difficult for existing buildings, as the soil on which the foundation is resting is not directly accessible for experiments. For the reasons stated above, the development of a method for the determination of *k* is an inverse problem of current interest in practice.

In this paper, we consider the stability issue in determining the Winkler subgrade coefficient of a slab foundation from the measurement of the deflection induced at interior points by a given load condition. The mechanical model is as follows. The slab foundation is described as a thin elastic plate with uniform thickness *h* and middle surface coinciding with a bounded Lipschitz domain *Ω*, a condition that occurs when the slab foundation is anchored to sufficiently rigid vertical walls. A concentrated force of intensity *f* is supposed to act at an internal point *P*_{0}∈*Ω*. This load condition has the merit of being easy to implement in practice. According to the Winkler model of soil and working in the framework of the Kirchhoff–Love theory of plates, the transversal displacement *w* of the plate satisfies the fourth-order Dirichlet boundary value problem
*n* is the unit outer normal to ∂*Ω*. Given the concentrated force *fδ*(*P*_{0}) and the coefficient *k*,

Here we are mainly interested in studying the stability of the unknown subgrade coefficient *k* in (1.1)–(1.3) from a single measurement of *w* inside *Ω*. It should be noted that the measurement of the transversal deflection at interior points of the plate can be easily carried out by means of no-contact techniques based on radar methodology [4].

Our main result states that, for *s*<1, satisfying a suitable structural condition (see (3.3)), if *i*=1,2, and if, for a given *ϵ*>0,
*σ*>0, we have
*Ω*_{σ}={*x*∈*Ω* | dist(*x*,∂*Ω*)>*σ*} and the constants *C*>0, *β*∈(0,1) only depend on the *a priori* data and on *σ*.

It should be noted that one difficulty of the problem comes from the fact that the displacement *w* may change sign and vanish somewhere inside *Ω*. See, for instance, the examples in Garabedian [5], Kozlov *et al.* [6] and Shapiro & Tegmark [7]. Therefore, it is necessary to keep under control the possible vanishing rate of *w*. Thus, the key ingredients of the proof are quantitative versions of the unique continuation principle for the solutions to the equation *A*_{p} property (proposition 3.5). Another useful tool is a pointwise lower bound in a neighbourhood of the point *P*_{0}, where the force is acting for solutions to (1.1) (lemma 3.3).

Let us mention that this method, essentially based on quantitative estimates of unique continuation, has some similarities, although with a different underlying equation and with a different kind of data, to the one used by Alessandrini [8] for another inverse problem with interior measurements arising in hybrid imaging.

The paper is organized as follows. Section 2 contains the notation, the formulation of the direct problem and a regularity result in fractional Sobolev spaces (proposition 2.3). Section 3 is devoted to the formulation and analysis of the inverse problem.

## 2. The direct problem

### (a) Notation

We shall denote by *B*_{r}(*P*) the open disc in *r* and centre *P*.

For any *r*>0, we denote

### Definition 2.1 (C^{k,α} regularity)

Let *Ω* be a bounded domain in *k*,*α*, with *k*=0,1,2,…, 0<*α*≤1, we say that a portion *S* of ∂*Ω* is of class *C*^{k,α} with constants *ρ*_{0}, *M*_{0}>0, if, for any *P*∈*S*, there exists a rigid transformation of coordinates under which we have *P*=*O* and
*ψ* is a *C*^{k,α} function defined in *I*_{ρ0}=(−*ρ*_{0},*ρ*_{0}) satisfying
*k*=0, *α*=1, we also say that *S* is of *Lipschitz class with constants ρ_{0}, M_{0}*.

We use the convention to normalize all norms in such a way that their terms are dimensionally homogeneous with the argument of the norm and coincide with the standard definition when the dimensional parameter equals one. For instance, the norm appearing above is meant as follows:
*ψ*^{(i)} denotes the *i*-order derivative of *ψ*.

Similarly, given a function *Ω* satisfies definition 2.1, and denoting by ∇^{i}*u* the vector whose components are the derivatives of order *i* of the function *u*, the norm above is denoted as

Moreover, for *k*=0,1,2,…, and *s*∈(0,1), we denote
_{s} is given by

For every 2×2 matrix *A*,*B* and every *A*^{T} the transpose of the matrix *A*.

### (b) Formulation of the direct problem

Let *ρ*_{0}, *M*_{0} and assume that
*Ω*×[−*h*/2,*h*/2] with middle surface represented by *Ω* and whose thickness *h* is much smaller than the characteristic dimension of *Ω*, that is, *h*≪*ρ*_{0}. The plate is made by linearly elastic material with elasticity tensor *C*_{αβγδ} satisfying the symmetry conditions
*A*,*B*, and the strong convexity condition
*A*, where *ξ*_{0}, *ξ*_{1} are positive constants.

The plate is resting on a Winkler soil with subgrade reaction coefficient
*Ω* is clamped and we assume that a concentrated force is acting at a point *P*_{0}∈*Ω* along a direction orthogonal to the middle surface *Ω*. According to the Kirchhoff–Love theory of thin plates subject to infinitesimal deformation, the statical equilibrium of the plate is described by the following Dirichlet boundary value problem:
*k* satisfies
*w*=*w*(*x*) is the transversal displacement at the point *x*∈*Ω* and *n* is the unit outer normal to ∂*Ω*.

We note that the presence in (2.11) and (2.15) of the parameter *ρ*_{0} (which has the dimension of a length) allows for a scaling-invariant formulation of the plate equation.

### Proposition 2.2

*Under the above assumptions, there exists a unique weak solution* *to* (*2.11*)–(*2.13*), *which satisfies*
*where the constant C*>0 *only depends on h*, *M*_{0}, *M*_{1} *and ξ*_{0}.

### Proof.

The weak formulation of problems (2.11)–(2.13) consists in finding *v*=*w* in (2.18), by (2.9) and using Poincaré inequality, we have
*C*>0 only depends on *h*, *M*_{0}, *M*_{1} and *ξ*_{0}. By (2.20) and the embedding (2.19), the thesis follows. ▪

In the analysis of the inverse problem, we shall need the following regularity result when the coefficients of the plate operator belong to a fractional Sobolev space.

### Proposition 2.3 (*H*^{s}-regularity)

*Let Ω be a bounded domain in* *with boundary of Lipschitz class with constants ρ*_{0}, *M*_{0}, *satisfying* (2.6). *Given g*∈*H*^{s}(*Ω*), *let w*∈*H*^{2}(*Ω*) *be a solution to*
*where* *is given by* (2.14), *with* *satisfying* (*2.7*)–(*2.9*) *and, for some s*∈(0,1),
*σ*>0, we have
*C*>0 only depends on *h*, *M*_{0}, *M*_{1}, *M*_{2}, *M*_{3}, *ξ*_{0}, *s* and *σ*.

### Proof.

When *M*_{2} and *M*_{3}. We omit the details. ▪

## 3. The inverse problem

In order to derive our stability result for the inverse problem, we need further *a priori* information.

Concerning the point *P*_{0} of the plate in which the concentrated force is acting, we assume that
*d*. On the elasticity tensor *a*_{0}=*C*_{1111}, *a*_{1}=4*C*_{1112}, *a*_{2}=2*C*_{1122}+4*C*_{1212}, *a*_{3}=4*C*_{2212}, *a*_{4}=*C*_{2222} and by *S*(*x*) the matrix is as follows:
*k*, we require the additional regularity

### Remark 3.1

Let us emphasize that the assumption *k*∈*H*^{s}(*Ω*) is not merely a mathematical technicality, but it can be grounded on realistic mechanical considerations. If, for instance, *k* is piecewise constant and is represented as
*E*_{1},…,*E*_{J} is a partition of *Ω* into disjoint subsets of finite perimeter (in the sense of Caccioppoli, i.e. *j*), then *k* belongs to *s*, *C*_{s} only depends on *k*. For a proof, see [12], formula (2.15) and also [13] for the convergence properties of the mollifications of *BV* functions.

In particular, if *k* is of the form (3.6) and we assume

for a given

Hereinafter, we shall refer to *h*, *d*, *M*_{0}, *M*_{1}, *M*_{2}, *M*_{3}, *ξ*_{0}, *s* as the *a priori data*.

### Theorem 3.2

*Let Ω be a bounded domain in* *with boundary of Lipschitz class with constants ρ*_{0}*, M*_{0}*, satisfying (2.6). Let* *be given by (2.14), with* *satisfying (2.7)–(2.9), (2.22), (2.23) for some s∈(0,1), and (3.3). Let P*_{0}*∈Ω satisfying (3.1).*

*Given f>0, let* *i=1,2, be the solution to
**for* *satisfying (2.15) and (3.5).*

*If, for some ϵ>0,
**then, for every σ>0, we have
**where the constants C>0 and β∈(0,1) only depend on the a priori data and on σ.*

As is obvious, the above stability result also implies uniqueness. Indeed, by the following arguments, it is easily seen that, under the above-stated structural conditions on

Let us premise to the proof of theorem 3.2 some auxiliary propositions concerning quantitative versions of the unique continuation principle (lemma 3.3 and propositions 3.4 and 3.5).

### Lemma 3.3

*Let Ω be a bounded domain with boundary* ∂*Ω of Lipschitz class with constants ρ*_{0}, *M*_{0}, *satisfying* (*2.6*). *Let P*_{0}∈*Ω satisfying* (*3.1*). *Let* *be given by* (*2.14*), *with* *satisfying* (*2.7*)–(*2.9*), *and let k and f satisfy* (*2.15*), (*2.16*), *respectively. Let* *be the solution to* (*2.11*)–(*2.13*). *There exists* *only depending on h, d, M*_{0}, *M*_{1}, *ξ*_{0} *and ξ*_{1}, *such that*
*where C*>0 *only depends on h*, *M*_{0}, *M*_{1}, *ξ*_{0} *and ξ*_{1},
*where C*>0 *only depends on h*, *M*_{0}, *M*_{1}, *ξ*_{0}, *ξ*_{1} *and*

### Proof.

By (2.9) and (2.18), we have that for every *C*>0 only depends on *h*, *ξ*_{1} and *B*_{dρ0}(*P*_{0})⊂*Ω* by (3.1), we have
*C*>0 is an absolute constant. From (3.15) and (3.16), it follows that
*C*>0 only depends on *h*, *ξ*_{1}, *C*>0 only depends on *h*, *M*_{0}, *M*_{1} and *ξ*_{0}. By (3.17), (3.18) and by the embedding inequality (2.19), we have
*C*>0 and *c*_{0}>0 only depend on *h*, *M*_{0}, *M*_{1}, *ξ*_{0}, *ξ*_{1} and *B*_{2σρ0}(*P*_{0})∖*B*_{σρ0}(*P*_{0}), which is contained in

### Proposition 3.4 (Lipschitz propagation of smallness)

*Let U be a bounded Lipschitz domain of* *with constants ρ*_{0}, *M*_{0} *and satisfying* *Let w*∈*H*^{2}(*U*) *be a solution to*
*where* *defined in* (*2.14*), *satisfies* (*2.7*)–(*2.9*) *and* (*2.22*) *in U, and k satisfies* (*2.15*) *in U. Assume*
*There exists a constant c*_{1}>1, *only depending on h, M*_{2}, *ξ*_{0} *and* *such that, for every τ*>0 *and for every x*∈*U*_{c1τρ0}, *we have*
*where c*_{τ}>0 *only depends on h*, *M*_{0}, *M*_{1}, *M*_{2}, *ξ*_{0}, *τ and on N*.

The proof of the above proposition is based on the three spheres inequality obtained by Lin *et al.* [14].

### Proposition 3.5 (*A*_{p} property)

*In the same hypotheses of proposition 3.4, there exists a constant c*_{2}>1, *only depending on h*, *M*_{0}, *M*_{1}, *M*_{2}, *ξ*_{0}, *such that, for every τ*>0 *and for every x*∈*U*_{c2τρ0}, *we have*
*where B*>0 *and p*>1 *only depend on h*, *M*_{0}, *M*_{1}, *M*_{2}, *ξ*_{0}, *τ and on N*.

The proof of the above proposition follows from the doubling inequality obtained by Di Cristo *et al.* [15] by applying the arguments in Garofalo & Lin [16].

### Proof of theorem 3.2

If *ϵ*≥1, then the proof of (3.11) is trivial in view of (2.15). Therefore, we restrict the analysis to the case 0<*ϵ*<1.

The difference
*i*=1,2 satisfies the boundary-value problem
*C*>0 an absolute constant. Let *g*=(*k*_{2}−*k*_{1})*w*_{1}−*k*_{2}*w*. Note that, by (2.15), (2.17), (2.19) and (3.5),
*C*>0 only depends on *h*, *M*_{0}, *M*_{1} and *ξ*_{0}. By applying proposition 2.3, we have
*C*>0 only depending on *h*, *M*_{0}, *M*_{1}, *M*_{2}, *M*_{3}, *ξ*_{0}, *s* and *σ*. From the well-known interpolation inequality,
*C*>0 only depends on *h*, *M*_{0}, *M*_{1}, *M*_{2}, *M*_{3}, *ξ*_{0}, *s* and *σ*. From (3.30), (3.33), (3.34) and (3.38), it follows that
*C*>0 only depends on *h*, *M*_{0}, *M*_{1}, *M*_{2}, *M*_{3}, *ξ*_{0}, *s* and *σ*.

Let us first estimate |*k*_{2}−*k*_{1}| in a disc centred at *P*_{0}. Note that, by the choice of *w*=*w*_{1}, and by (3.39) with *C*>0 only depends on *h*, *M*_{0}, *M*_{1}, *M*_{2}, *M*_{3}, *ξ*_{0}, *s* and *d*.

Now, let us control |*k*_{2}−*k*_{1}| in
*A*_{p}-property and the Lipschitz propagation of smallness.

By applying the Hölder inequality and (3.39), we can write, for every *p*>1,
*C*>0 only depends on *h*, *M*_{0}, *M*_{1}, *M*_{2}, *M*_{3}, *ξ*_{0}, *s* and *σ*.

Let us cover *Q*_{l}(*x*_{j}) with centre *x*_{j} and side *j*=1,…,*J*, where *c*_{1} and *c*_{2} have been introduced in proposition 3.4 and in proposition 3.5, respectively. By the choice of *l*, denoting *A*_{p}-property (3.25) and the Lipschitz propagation of smallness property (3.24) to *w*=*w*_{1} in *j*, *j*=1,…,*J*, dist(*x*_{j},∂*U*)≥*c*_{i}*r*, *i*=1,2, we have
*B*>0, *p*>1 and *c*_{τ}>0 only depend on *h*, *M*_{0}, *M*_{1}, *M*_{2}, *ξ*_{0}, *σ* and on the frequency ratio *C*>0 only depends on *h*, *M*_{0}, *M*_{1}, *ξ*_{0}, *ξ*_{1} and *C*>0 only depends on *h*, *M*_{0}, *M*_{1}, *M*_{2}, *ξ*_{0}, *d* and *σ*. By (3.42) and (3.47), we have
*C*>0 only depends on *h*, *M*_{0}, *M*_{1}, *M*_{2}, *M*_{3}, *ξ*_{0}, *s*, *d* and *σ*. Finally, by (3.40) and (3.48), and recalling that *p*>1 and *ϵ*<1, estimate (3.11) follows with *β*=*s*/(*p*(4+*s*)). ▪

## 4. Conclusion

In this paper, we have shown, by means of a rigorous mathematical analysis, that the nonlinear inverse problem of determining the Winkler coefficient *k* from the measurement of the transversal displacement *w* induced by a load concentrated at one point is only mildly ill-posed.

As is well known [17,18], the Hölder stability estimates we achieved imply convergence of regularized inversion procedures. It can also be noted that, for this specific problem, although it is nonlinear, the inversion can be performed by a cascade of linear inversion procedures. Namely,

(i) from

*w*obtain*kw*; and(ii) from

*kw*obtain*k*.

This approach shall be the object of a subsequent study. We emphasize that, from such regularized inversion procedures, it will be possible to test the efficiency of our proposed method with the aid of indoor and field experiments already available in the civil engineering literature.

On the other hand, we are aware that also multiple concentrated loads and distributed loads are of great relevance in civil engineering, and also that different models of foundations, other than the Kirchhoff–Love one, are of interest. However, it is reasonable to expect that, under such different modelling assumptions, a different mathematical analysis shall be needed.

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## Authors' contributions

All authors contributed equally to the research. All authors gave final approval for publication.

## Competing interests

We have no competing interests.

## Funding

G.A. and E.R. are supported by FRA2014 ‘Problemi inversi per PDE, unicità, stabilità, algoritmi’, Università degli Studi di Trieste; A.M. has been partially supported by the Carlos III University of Madrid-Banco de Santander Chairs of Excellence Programme for the 2013–2014 Academic Year; and E.R. and S.V. are partially supported by GNAMPA of Istituto Nazionale di Alta Matematica.

## Acknowledgements

The authors thank the anonymous referees for providing stimulating suggestions on possible extensions and practical applications of our stability estimates, as well as for pointing out useful bibliographical information.

- Received February 20, 2015.
- Accepted May 22, 2015.

- © 2015 The Author(s) Published by the Royal Society. All rights reserved.